Derive
Using
Formula
Mean Residual Life Function
PDF
MeanResidualLifeFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{x}^{\infty }(t-x){\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}}}
Mean Residual Life Function
PDF
MeanResidualLifeFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{0}^{\infty }t{\text{PDF}}[{\mathcal {D}},x+t]\,dt}{\int _{0}^{\infty }{\text{PDF}}[{\mathcal {D}},x+t]\,dt}}}
Mean Residual Life Function
PDF
MeanResidualLifeFunction
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∞
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}}}
Mean Residual Life Function
PDF
MeanResidualLifeFunction
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Mean
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x-\int _{-\infty }^{x}(t-x){\text{PDF}}[{\mathcal {D}},t]\,dt}{1-\int _{-\infty }^{x}{\text{PDF}}[{\mathcal {D}},t]\,dt}}}
Mean Residual Life Function
PDF
MeanResidualLifeFunction
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=
(
{
Mean
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Mean
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v
Indeterminate
True
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/;
u
=
support
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1
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1
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∧
v
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support
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=\left({\begin{array}{cc}\{&{\begin{array}{cc}{\text{Mean}}[{\mathcal {D}}]-x&x<u\\-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{u}^{x}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}}&u\leq x<v\\{\text{Indeterminate}}&{\text{True}}\\\end{array}}\\\end{array}}\right){\text{/;}}u={\text{support}}[{\mathcal {D}}][[1,1]]\land v={\text{support}}[{\mathcal {D}}][[1,2]]}
Mean Residual Life Function
CDF
MeanResidualLifeFunction
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Mean
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x
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CDF
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d
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CDF
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x+\int _{-\infty }^{x}{\text{CDF}}[{\mathcal {D}},t]\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
CDF
MeanResidualLifeFunction
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CDF
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t
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CDF
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{x}^{\infty }(t-x){\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
CDF
MeanResidualLifeFunction
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=
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CDF
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∂
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t
1
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CDF
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{0}^{\infty }t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t+x]}{\partial t}}\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
CDF
MeanResidualLifeFunction
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∂
CDF
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∂
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d
t
1
−
CDF
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Characteristic Function
MeanResidualLifeFunction
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CharacteristicFunction
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π
SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Characteristic Function
MeanResidualLifeFunction
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2
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τ
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CharacteristicFunction
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d
t
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π
−
(
2
i
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τ
[
CharacteristicFunction
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τ
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x
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-x)\right)}}}
Mean Residual Life Function
Survival Function
MeanResidualLifeFunction
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Mean
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SurvivalFunction
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SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x+\int _{-\infty }^{x}(1-{\text{SurvivalFunction}}[{\mathcal {D}},t])\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Survival Function
MeanResidualLifeFunction
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SurvivalFunction
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SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-{\frac {\int _{x}^{\infty }(t-x){\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Survival Function
MeanResidualLifeFunction
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SurvivalFunction
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SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-{\frac {\int _{0}^{\infty }t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t+x]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Survival Function
MeanResidualLifeFunction
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∂
SurvivalFunction
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∂
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t
SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x-{\frac {\int _{x}^{\infty }t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Survival Function
MeanResidualLifeFunction
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Mean
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SurvivalFunction
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∂
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t
SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]+\int _{-\infty }^{x}t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Cumulative Hazard Function
MeanResidualLifeFunction
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CumulativeHazardFunction
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∂
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CumulativeHazardFunction
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SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-{\frac {\int _{0}^{\infty }{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t+x)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t+x)}}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Cumulative Hazard Function
MeanResidualLifeFunction
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x
exp
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CumulativeHazardFunction
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Mean
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CumulativeHazardFunction
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∂
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e
CumulativeHazardFunction
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SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},x))+{\text{Mean}}[{\mathcal {D}}]-\int _{-\infty }^{x}{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Hazard Function
MeanResidualLifeFunction
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(
t
exp
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HazardFunction
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HazardFunction
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t
SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{0}^{\infty }\left(t\exp \left(-\int _{-\infty }^{t+x}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right){\text{HazardFunction}}[{\mathcal {D}},t+x]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Hazard Function
MeanResidualLifeFunction
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D
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x
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=
−
x
exp
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−
∫
−
∞
x
HazardFunction
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t
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Mean
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∫
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x
(
t
exp
(
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∫
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∞
t
HazardFunction
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d
τ
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)
HazardFunction
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d
t
SurvivalFunction
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x
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x\exp \left(-\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\,dt\right)+{\text{Mean}}[{\mathcal {D}}]-\int _{-\infty }^{x}\left(t\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right){\text{HazardFunction}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Inverse CDF
MeanResidualLifeFunction
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−
∞
∞
∫
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∞
∫
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1
t
e
(
i
(
InverseCDF
[
D
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−
t
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x
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ξ
d
τ
d
ξ
d
t
(
2
π
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SurvivalFunction
[
D
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x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{InverseCDF}}[{\mathcal {D}},\tau ]-t-x))\xi }d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Inverse CDF
MeanResidualLifeFunction
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−
x
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Mean
[
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−
∫
0
CDF
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InverseCDF
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d
t
SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Quantile
MeanResidualLifeFunction
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e
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i
(
Quantile
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ξ
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τ
d
ξ
d
t
2
π
SurvivalFunction
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x
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{Quantile}}[{\mathcal {D}},\tau ]-t))\xi }d\tau d\xi dt}{2\pi }}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Quantile
MeanResidualLifeFunction
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∞
∫
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e
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i
(
Quantile
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−
t
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ξ
d
τ
d
ξ
d
t
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2
π
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SurvivalFunction
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D
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x
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{Quantile}}[{\mathcal {D}},\tau ]-t-x))\xi }d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Quantile
MeanResidualLifeFunction
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=
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x
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Mean
[
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−
∫
0
CDF
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Quantile
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d
t
SurvivalFunction
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x
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}{\text{Quantile}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Inverse Survival Function
MeanResidualLifeFunction
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=
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−
∞
∞
∫
−
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∞
∫
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t
e
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i
(
InverseSurvivalFunction
[
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t
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ξ
d
τ
d
ξ
d
t
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π
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SurvivalFunction
[
D
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x
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-t-x))\xi }d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Inverse Survival Function
MeanResidualLifeFunction
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=
−
x
+
Mean
[
D
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−
∫
SurvivalFunction
[
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1
InverseSurvivalFunction
[
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d
t
SurvivalFunction
[
D
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x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{{\text{SurvivalFunction}}[{\mathcal {D}},x]}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Sparsity
MeanResidualLifeFunction
(
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x
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=
∫
−
∞
∞
∫
−
∞
∞
∫
0
1
t
exp
(
(
i
(
Median
[
D
]
+
∫
1
2
τ
Sparsity
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d
s
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ξ
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d
τ
d
ξ
d
t
(
2
π
)
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}t\exp \left(\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},s)\,ds-t-x\right)\right)\xi \right)d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Sparsity
MeanResidualLifeFunction
(
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x
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=
−
x
(
1
−
CDF
[
D
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x
]
)
+
Mean
[
D
]
−
∫
−
∞
x
∫
−
∞
∞
∫
0
1
t
exp
(
(
i
(
Median
[
D
]
+
∫
1
2
τ
Sparsity
(
D
,
s
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d
s
−
t
)
)
ξ
)
d
τ
d
ξ
d
t
2
π
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x(1-{\text{CDF}}[{\mathcal {D}},x])+{\text{Mean}}[{\mathcal {D}}]-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}t\exp \left(\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},s)\,ds-t\right)\right)\xi \right)d\tau d\xi dt}{2\pi }}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Sparsity
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
(
1
−
CDF
[
D
,
x
]
)
+
Mean
[
D
]
−
∫
0
CDF
[
D
,
x
]
(
Median
[
D
]
+
∫
1
2
t
Sparsity
(
D
,
s
)
d
s
)
d
t
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x(1-{\text{CDF}}[{\mathcal {D}},x])+{\text{Mean}}[{\mathcal {D}}]-\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Lorenz Curve 1
MeanResidualLifeFunction
(
D
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x
)
=
Mean
[
D
]
(
1
−
LorenzCurve1
(
D
,
x
)
)
SurvivalFunction
[
D
,
x
]
−
x
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-{\text{LorenzCurve1}}({\mathcal {D}},x))}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}-x}
Mean Residual Life Function
Lorenz Curve 1
MeanResidualLifeFunction
(
D
,
x
)
=
Mean
[
D
]
∫
0
∞
t
∂
LorenzCurve1
(
D
,
t
+
x
)
∂
t
t
+
x
d
t
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]\int _{0}^{\infty }{\frac {t{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t+x)}{\partial t}}}{t+x}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Lorenz Curve 1
MeanResidualLifeFunction
(
D
,
x
)
=
Mean
[
D
]
(
1
−
LorenzCurve1
(
D
,
x
)
)
−
x
+
(
x
Mean
[
D
]
)
∫
−
∞
x
∂
LorenzCurve1
(
D
,
t
)
∂
t
t
d
t
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-{\text{LorenzCurve1}}({\mathcal {D}},x))-x+(x{\text{Mean}}[{\mathcal {D}}])\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Lorenz Curve 1
MeanResidualLifeFunction
(
D
,
x
)
=
Mean
[
D
]
−
x
+
(
Mean
[
D
]
x
)
∫
−
∞
x
LorenzCurve1
(
D
,
t
)
t
2
d
t
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x+({\text{Mean}}[{\mathcal {D}}]x)\int _{-\infty }^{x}{\frac {{\text{LorenzCurve1}}({\mathcal {D}},t)}{t^{2}}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Lorenz Curve 2
MeanResidualLifeFunction
(
D
,
x
)
=
Mean
[
D
]
(
1
−
LorenzCurve2
(
D
,
CDF
[
D
,
x
]
)
)
SurvivalFunction
[
D
,
x
]
−
x
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x]))}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}-x}
Mean Residual Life Function
Lorenz Curve 2
MeanResidualLifeFunction
(
D
,
x
)
=
Mean
[
D
]
∫
0
∞
t
∂
LorenzCurve2
(
D
,
CDF
[
D
,
t
+
x
]
)
∂
t
t
+
x
d
t
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]\int _{0}^{\infty }{\frac {t{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t+x])}{\partial t}}}{t+x}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Mean Residual Life Function
identical properties
{\displaystyle {\text{identical properties}}}
Mean Residual Life Function
Moment
MeanResidualLifeFunction
(
D
,
x
)
=
Mean
[
D
]
(
1
−
θ
(
x
)
)
−
x
θ
(
−
x
)
+
∑
k
=
0
∞
(
(
−
1
)
k
−
1
δ
(
k
)
(
x
)
)
(
x
Moment
[
D
,
k
+
1
]
+
Moment
[
D
,
k
+
2
]
)
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-\theta (x))-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)(x{\text{Moment}}[{\mathcal {D}},k+1]+{\text{Moment}}[{\mathcal {D}},k+2])}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Moment
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
+
θ
(
−
x
)
Mean
[
D
]
+
∑
k
=
0
∞
(
(
−
1
)
k
δ
(
k
)
(
x
)
)
Moment
[
D
,
k
+
2
]
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k+2]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Central Moment
MeanResidualLifeFunction
(
D
,
x
)
=
Mean
[
D
]
(
1
−
θ
(
x
)
)
−
x
θ
(
−
x
)
+
∑
k
=
0
∞
(
(
−
1
)
k
−
1
δ
(
k
)
(
x
)
)
(
x
∑
j
=
0
k
+
1
(
(
k
+
1
j
)
Mean
[
D
]
k
+
1
−
j
)
CentralMoment
[
D
,
j
]
+
∑
j
=
0
k
+
2
(
(
k
+
2
j
)
Mean
[
D
]
k
+
2
−
j
)
CentralMoment
[
D
,
j
]
)
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-\theta (x))-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+1-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]+\sum _{j=0}^{k+2}\left({\binom {k+2}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+2-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Central Moment
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
+
θ
(
−
x
)
Mean
[
D
]
+
∑
k
=
0
∞
(
(
−
1
)
k
δ
(
k
)
(
x
)
)
∑
j
=
0
k
+
2
(
(
k
+
2
j
)
Mean
[
D
]
k
+
2
−
j
)
CentralMoment
[
D
,
j
]
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+2}\left({\binom {k+2}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+2-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Factorial Moment
MeanResidualLifeFunction
(
D
,
x
)
=
Mean
[
D
]
(
1
−
θ
(
x
)
)
−
x
θ
(
−
x
)
+
∑
k
=
0
∞
(
(
−
1
)
k
−
1
δ
(
k
)
(
x
)
)
(
x
∑
j
=
0
k
+
1
S
k
+
1
(
j
)
FactorialMoment
[
D
,
j
]
+
∑
j
=
0
k
+
2
S
k
+
2
(
j
)
FactorialMoment
[
D
,
j
]
)
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-\theta (x))-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]+\sum _{j=0}^{k+2}{\mathcal {S}}_{k+2}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Factorial Moment
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
+
θ
(
−
x
)
Mean
[
D
]
+
∑
k
=
0
∞
(
(
−
1
)
k
δ
(
k
)
(
x
)
)
∑
j
=
0
k
+
2
S
k
+
2
(
j
)
FactorialMoment
[
D
,
j
]
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+2}{\mathcal {S}}_{k+2}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Ascending Factorial Moment
MeanResidualLifeFunction
(
D
,
x
)
=
Mean
[
D
]
(
1
−
θ
(
x
)
)
−
x
θ
(
−
x
)
+
∑
k
=
0
∞
(
(
−
1
)
k
−
1
δ
(
k
)
(
x
)
)
(
x
∑
j
=
0
k
+
1
(
(
−
1
)
k
−
j
−
1
S
k
+
1
(
j
)
)
AscendingFactorialMoment
(
D
,
j
)
+
∑
j
=
0
k
+
2
(
(
−
1
)
k
−
j
S
k
+
2
(
j
)
)
AscendingFactorialMoment
(
D
,
j
)
)
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-\theta (x))-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x\sum _{j=0}^{k+1}\left((-1)^{k-j-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)+\sum _{j=0}^{k+2}\left((-1)^{k-j}{\mathcal {S}}_{k+2}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Ascending Factorial Moment
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
+
θ
(
−
x
)
Mean
[
D
]
+
∑
k
=
0
∞
(
(
−
1
)
k
δ
(
k
)
(
x
)
)
∑
j
=
0
k
+
2
(
(
−
1
)
k
−
j
S
k
+
2
(
j
)
)
AscendingFactorialMoment
(
D
,
j
)
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+2}\left((-1)^{k-j}{\mathcal {S}}_{k+2}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Cumulant
MeanResidualLifeFunction
(
D
,
x
)
=
∑
k
=
0
∞
(
(
−
1
)
k
−
1
δ
(
k
)
(
x
)
)
(
BellY
[
k
+
2
,
1
,
Table
[
{
1
,
Cumulant
[
D
,
j
]
}
,
{
j
,
k
+
2
}
]
]
+
x
BellY
[
k
+
1
,
1
,
Table
[
{
1
,
Cumulant
[
D
,
j
]
}
,
{
j
,
k
+
1
}
]
]
)
(
k
+
1
)
!
+
(
1
−
θ
(
x
)
)
Mean
[
D
]
−
x
θ
(
−
x
)
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)({\text{BellY}}[k+2,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+2\}]]+x{\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]])}{(k+1)!}}+(1-\theta (x)){\text{Mean}}[{\mathcal {D}}]-x\theta (-x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Cumulant
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
+
θ
(
−
x
)
Mean
[
D
]
+
∑
k
=
0
∞
(
(
−
1
)
k
δ
(
k
)
(
x
)
)
BellY
[
k
+
2
,
1
,
Table
[
{
1
,
Cumulant
[
D
,
j
]
}
,
{
j
,
k
+
2
}
]
]
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}[k+2,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+2\}]]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Factorial Cumulant
MeanResidualLifeFunction
(
D
,
x
)
=
Mean
[
D
]
(
1
−
θ
(
x
)
)
−
x
θ
(
−
x
)
+
∑
k
=
0
∞
(
(
−
1
)
k
−
1
δ
(
k
)
(
x
)
)
(
x
BellY
[
k
+
1
,
1
,
Table
[
{
1
,
∑
i
=
1
j
S
j
(
i
)
FactorialCumulant
(
D
,
i
)
}
,
{
j
,
k
+
1
}
]
]
+
BellY
[
k
+
2
,
1
,
Table
[
{
1
,
∑
i
=
1
j
S
j
(
i
)
FactorialCumulant
(
D
,
i
)
}
,
{
j
,
k
+
2
}
]
]
)
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}](1-\theta (x))-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x{\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]+{\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Factorial Cumulant
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
+
θ
(
−
x
)
Mean
[
D
]
+
∑
k
=
0
∞
(
(
−
1
)
k
δ
(
k
)
(
x
)
)
BellY
[
k
+
2
,
1
,
Table
[
{
1
,
∑
i
=
1
j
S
j
(
i
)
FactorialCumulant
(
D
,
i
)
}
,
{
j
,
k
+
2
}
]
]
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Ascending Factorial Cumulant
MeanResidualLifeFunction
(
D
,
x
)
=
∑
k
=
0
∞
(
(
−
1
)
k
−
1
δ
(
k
)
(
x
)
)
(
BellY
[
k
+
2
,
1
,
Table
[
{
1
,
∑
i
=
0
j
(
(
−
1
)
j
−
i
S
j
(
i
)
)
AscendingFactorialCumulant
(
D
,
i
)
}
,
{
j
,
k
+
2
}
]
]
+
x
BellY
[
k
+
1
,
1
,
Table
[
{
1
,
∑
i
=
0
j
(
(
−
1
)
j
−
i
S
j
(
i
)
)
AscendingFactorialCumulant
(
D
,
i
)
}
,
{
j
,
k
+
1
}
]
]
)
(
k
+
1
)
!
+
(
1
−
θ
(
x
)
)
Mean
[
D
]
−
x
θ
(
−
x
)
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left({\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]+x{\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]\right)}{(k+1)!}}+(1-\theta (x)){\text{Mean}}[{\mathcal {D}}]-x\theta (-x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Ascending Factorial Cumulant
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
+
θ
(
−
x
)
Mean
[
D
]
+
∑
k
=
0
∞
(
(
−
1
)
k
δ
(
k
)
(
x
)
)
BellY
[
k
+
2
,
1
,
Table
[
{
1
,
∑
i
=
0
j
(
(
−
1
)
j
−
i
S
j
(
i
)
)
AscendingFactorialCumulant
(
D
,
i
)
}
,
{
j
,
k
+
2
}
]
]
(
k
+
1
)
!
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Moment Generating Function
MeanResidualLifeFunction
(
D
,
x
)
=
1
2
(
Mean
[
D
]
−
x
)
−
F
τ
[
MomentGeneratingFunction
[
D
,
i
τ
]
τ
2
]
(
−
x
)
2
π
SurvivalFunction
[
D
,
x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\frac {1}{2}}({\text{Mean}}[{\mathcal {D}}]-x)-{\frac {{\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau ^{2}}}\right](-x)}{\sqrt {2\pi }}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Moment Generating Function
MeanResidualLifeFunction
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MomentGeneratingFunction
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SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Moment Generating Function
MeanResidualLifeFunction
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MomentGeneratingFunction
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MomentGeneratingFunction
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τ
]
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−
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)
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}}}
Mean Residual Life Function
Central Moment Generating Function
MeanResidualLifeFunction
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2
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Mean
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Mean
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CentralMomentGeneratingFunction
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π
SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\frac {1}{2}}({\text{Mean}}[{\mathcal {D}}]-x)-{\frac {{\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau ^{2}}}\right](-x)}{\sqrt {2\pi }}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Central Moment Generating Function
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
+
∫
x
∞
t
(
F
τ
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e
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Mean
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CentralMomentGeneratingFunction
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]
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SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Central Moment Generating Function
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
+
2
∫
x
∞
t
(
F
τ
[
e
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Mean
[
D
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CentralMomentGeneratingFunction
[
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]
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π
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2
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τ
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e
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i
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Mean
[
D
]
CentralMomentGeneratingFunction
[
D
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i
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τ
]
(
−
x
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)
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}}}
Mean Residual Life Function
Cumulant Generating Function
MeanResidualLifeFunction
(
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)
=
1
2
(
Mean
[
D
]
−
x
)
−
F
τ
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exp
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CumulantGeneratingFunction
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)
τ
2
]
(
−
x
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2
π
SurvivalFunction
[
D
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x
]
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\frac {1}{2}}({\text{Mean}}[{\mathcal {D}}]-x)-{\frac {{\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau ^{2}}}\right](-x)}{\sqrt {2\pi }}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Cumulant Generating Function
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
+
∫
x
∞
t
(
F
τ
[
exp
(
CumulantGeneratingFunction
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)
]
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SurvivalFunction
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{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}
Mean Residual Life Function
Cumulant Generating Function
MeanResidualLifeFunction
(
D
,
x
)
=
−
x
+
2
∫
x
∞
t
(
F
τ
[
exp
(
CumulantGeneratingFunction
[
D
,
i
τ
]
)
]
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−
t
)
)
d
t
2
π
−
(
2
i
)
(
F
τ
[
exp
(
CumulantGeneratingFunction
[
D
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i
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)
τ
]
(
−
x
)
)
{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-x)\right)}}}
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\partial {\text{CDF}}[{\mathcal {D}},x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1504b26e9b80e8ec34c136a57a2049a5fe12fc0c)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},t]}{e^{(it)x}}}\,dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/627c8281354d7dda39b93db1d304452b66f6c8f7)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {{\mathcal {F}}_{t}[{\text{CharacteristicFunction}}[{\mathcal {D}},t]](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ca07cb8b66e1dfd7044e7380a7e6f5f6d3dab6)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=-{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4cd94479a19381f8521540d8068a273b858632)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},x)}{\partial x}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},x)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e849dd224ca9d78d36338fb458cebf21c9464c7)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\text{HazardFunction}}[{\mathcal {D}},x]\exp \left(-\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\,dt\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/977881612a366cba975fc4bba74c597717a8a96f)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b9402ade0d9ff309632527b74a43da77435f9f9)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x)\,d\tau }{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91bc3081fb3c57b44be0de0afb9a80ff5b14b28e)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=\int _{0}^{1}\delta ({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x)\,d\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/858f49a3ce28e02e4e8b02f69f2b3976aa0dd58e)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{Quantile}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53fe3073b125609da7772efc1d2cf6eb4980a57d)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{Quantile}}[{\mathcal {D}},\tau ]-x)\,d\tau }{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45021e99d7e1a686562c4739f5e4aba34e7de831)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=\int _{0}^{1}\delta ({\text{Quantile}}[{\mathcal {D}},\tau ]-x)\,d\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/51ceff11256449caff4434f1dbe17714b1ca21bd)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ebc4362491a9babbe2633ee54c0ff259913e8c9)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x)\,d\tau }{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5cba4ee68c184b0cbd50bd46e05406f96046b13)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=\int _{0}^{1}\delta ({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x)\,d\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0560014fccb53bd7aaf3b4df438a73d79f45aba2)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\right)t\right)d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c76c4add3c827ca8f663512d29b472ff9d2ec853)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}{\mathcal {F}}_{t}[1]\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\,d\tau }{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c110038c8f6b837e0b70d914c688e056682e48d4)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=\int _{0}^{1}\delta \left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\,d\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/db0f24829c6f5fe6da765368c2c3638246246261)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},x)}{\partial x}}}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/785ecca6750f897419be662f6ec229c005cc76bf)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{\partial x}}}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1beb54473154f505b174c4cd82d6583f9a683ddf)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},x){\text{SurvivalFunction}}[{\mathcal {D}},x])}{\partial x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7525e7fe89257920ec253d526ce0bb05f0a0645e)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k]}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a33f36bda3117f8c4bf9d1755a4ba9288429a82)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k}{j}}{\text{Mean}}[{\mathcal {D}}]^{k-j}\right)}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d161b8d90f4ede83bd6bf9f076d3d7a2280c8fc9)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k}{\mathcal {S}}_{k}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f14ebb65d3dc41ffbcc2ecc6b3b135681b9f4a34)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=\sum _{j=0}^{\infty }\left((-1)^{j}{\text{AscendingFactorialMoment}}({\mathcal {D}},j)\right)\sum _{k=j}^{\infty }{\frac {{\mathcal {S}}_{k}^{(j)}\delta ^{(k)}(x)}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d025c1f868a5bcc419467781a8cfd3edbde83a4)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}[k,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k\}]]}{k!}}+\delta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1874ec3cb9654c3f1a93e37f187229ea9ee127d0)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k\}\right]\right]}{k!}}+\delta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98e8c7504f282e07a79b2d909236547325554b62)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]=\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k\}\right]\right]}{k!}}+\delta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36bed213cb6e441b9c4158d3b3e3ad0cf9c593c8)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},t]}{e^{tx}}}\,dt}{(2\pi )i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a8f82ae1057ff0a10a469895020e6d1e1d2375e)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},it]}{e^{(it)x}}}\,dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/713063c838417b9087d11ea9e32ded676df52257)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {{\mathcal {F}}_{t}[{\text{MomentGeneratingFunction}}[{\mathcal {D}},it]](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2e24ab0b11d1a493573fca52ec6f722c1e0409)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-i\infty }^{i\infty }e^{t({\text{Mean}}[{\mathcal {D}}]-x)}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},t]\,dt}{(2\pi )i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/953a2924a8b29f6ad5733139040db0ed1465d047)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }e^{(it)({\text{Mean}}[{\mathcal {D}}]-x)}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},it]\,dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e460114cd157a572239d732068d48a81799540)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {{\mathcal {F}}_{t}\left[e^{(it){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},it]\right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d417730c438300e7cbfe6ab534f171de860cc93d)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},it]-(it)x}\,dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c34d70fbd188d5173579369e8625b8f3404a2e7)
![{\displaystyle {\text{PDF}}[{\mathcal {D}},x]={\frac {{\mathcal {F}}_{t}\left[e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},it]}\right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/288f436e5f1abb737a10f5bb5514ee3214476e2f)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=\int _{-\infty }^{x}{\text{PDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b059064b0d20173b589c304808ea7f67b910b6a6)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]{\text{PDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f51a88d05670cf7b97499c53c0914a340c13106c)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }\theta (x-t){\text{PDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50307c95e0ed1ed1f5d23e2e141e526b016eabba)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(it)\tau }}}d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcde4cb653b4300dc0f5a23a3fcd43d375fdf02b)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c73c18b2228ea6ee296871eaa97500c8d772d5f3)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5c904b5e24d8e37c2d5ba60f078b2d286e03d57)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\theta (x-t)\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f53a4042ec34e90ccd29375d65a74a2e702105b5)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b059eb7e2d9ef6688d7441d18a3151cb7c631d)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {\int _{0}^{\infty }{\frac {\Im \left({\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},t]}{e^{(it)x}}}\right)}{t}}\,dt}{\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/518990065dd165d352c5926aa4c90212331c377a)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=1-{\text{SurvivalFunction}}[{\mathcal {D}},x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e22f03a62a78473a19d09e55a5c47638aff0a3c)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac03717fd4a793f3f994c683938a5a322f12c0a)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=1-\exp \left(-\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\,dt\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e89f1be59847aa6f5e9dd69e723a38c5381e87)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{InverseCDF}}[{\mathcal {D}},t]][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cd0fd856c5eaf2c69066a0ec1f98b06aa558325)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{InverseCDF}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/974c60b7b07f24b5f942285019ab5cf16e80c2ab)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{Quantile}}[{\mathcal {D}},t]][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3895ae10b84af6f5d4cedd2a49f78a05900d38bf)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }{\frac {\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},\xi ])\tau }\,d\xi }{e^{(it)\tau }}}d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c160bb7339383b92d411ab94e982ce7a8248af0)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }\left[\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},\xi ])\tau }\,d\xi \right](-t)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a98a9d4016bae0c121ace67b84e1ef09ec3b380)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]\left({\mathcal {F}}_{\tau }\left[\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},\xi ])\tau }\,d\xi \right](-t)\right)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39df0769692d64ab9529556d2c336245950d2d80)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\theta (x-t)\left({\mathcal {F}}_{\tau }\left[\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},\xi ])\tau }\,d\xi \right](-t)\right)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc2be02af3decb1b51ea53dc1d02e73a4596f5a)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2359392c5f9f1d6c8e0c2edca8d97fadaa89f8d7)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{InverseSurvivalFunction}}[{\mathcal {D}},1-t]][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df3b9d8a39a46a3490ebd456fa365cbc0bb0e2bc)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/143ca241e68582adccb3957dafa2b9fa2ba12fa8)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\tau )\,d\tau \right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c6fbd28c83b80373b936ece299a53ee9100c778)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\right)\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e97ceacb8a51510373173735b23bfd85c775d646)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7db3783b79de10a85659283f413a08d3f4cce83b)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve1}}({\mathcal {D}},t)}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve1}}({\mathcal {D}},x)}{x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83cb692da73f2c014d5204d733a31c55497c021b)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}{t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63149f2941921b25d6f65c08e15811621bf23a96)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2bbce6623d76ae9b0814ccf9c4a01073068a7e)
![{\displaystyle \left({\begin{array}{c}{\text{SurvivalFunction}}[{\mathcal {D}},x]=\exp \left(-\int _{-\infty }^{x}{\frac {1+{\frac {\partial {\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}{\partial t}}}{{\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}}\,dt\right)\\{\text{CDF}}[{\mathcal {D}},x]=1+{\frac {\partial ({\text{SurvivalFunction}}[{\mathcal {D}},x]{\text{MeanResidualLifeFunction}}({\mathcal {D}},x))}{\partial x}}\\\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5ce1693defbb09ff49e226463036dcd932c9c5)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d6ab832bd0263135b9e973ea8394537db17546)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{-j+k+1}\right)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c711c0903bd5b9c82eabb7b357b6346d8e8d9d16)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/395177042597d18fe1ac7af7c18a63f8b899680e)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dd1d72a1e676e1f0ea9c68530eb2685ed7b1260)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30946a8d7482a5ebed820c6f97a6d9c4cea2243d)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edbcf59aeace67d7baef3757872fae2cf9081c6d)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(x)+1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42274b3cac3cdae5036dae1f64f91cc340dbae67)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},\tau ]}{e^{t\tau }}}d\tau dt}{(2\pi )i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eacba738a4e670a67aa2ca178912c4c5e62efe85)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6d2fca40d9ce042a4cb5e757fa368fbdfc8482)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/274371887f497d36d528af507073f4b273eb2971)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{(i({\text{Mean}}[{\mathcal {D}}]-t))\tau }{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f8f09c5419f882144f748f3c3c3772db07080e)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66b04e48ca47911988c1eca120cf9f36b0bd4f50)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf970c779e3222aaad0db9f1039422c9b6cb2834)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ]-(it)\tau }d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f8914fe8df7252d15ad0f0254527b4fdaafc44)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e20837cb81a23efd1b839d2200817705780a5fa7)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef9f001dbe7fa8ed1e79cb578fbdc79127fba54)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }e^{(it)x}{\text{PDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb6da17938182c9a684b641c2af762d43c3ca1c2)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]={\sqrt {2\pi }}\left({\mathcal {F}}_{t}[{\text{PDF}}[{\mathcal {D}},t]](x)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d00ec055fca84c9fe200b1c4020e5cdcb6c6848)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }e^{(it)x}{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2340dcfe0a14ea1041e8b2b4492b8638562bf1)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]={\sqrt {2\pi }}\left({\mathcal {F}}_{t}\left[{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\right](x)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a48f21bfdd75eff2be40508af75d5f4c70a30a)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=-\int _{-\infty }^{\infty }e^{(it)x}{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa506be9d2b851e1ee38450fe78e02ed266bd3a)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=-{\sqrt {2\pi }}\left({\mathcal {F}}_{t}\left[{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\right](x)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/467a418633ae3d9b8bca11a46de7841af5e22881)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}e^{-{\text{CumulativeHazardFunction}}({\mathcal {D}},t)+(it)x}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03147410b577525bf456cd4acac9e2d10fba77ae)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]={\sqrt {2\pi }}\left({\mathcal {F}}_{t}\left[{\frac {\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\right](x)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/034ebc577d9bd738b5dad6195961917557c4f497)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }\exp \left((it)x-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right){\text{HazardFunction}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c9165d7089a67a808462b8258698c1bad25309)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]={\sqrt {2\pi }}\left({\mathcal {F}}_{t}\left[{\text{HazardFunction}}[{\mathcal {D}},t]\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right](x)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d755b845d03b8e002a97062e0107a1ff8e56605)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{0}^{1}e^{x(i{\text{InverseCDF}}[{\mathcal {D}},t])}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c14b8be27321ac9ae949101cd38bbe702f424e)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{0}^{1}e^{x(i{\text{Quantile}}[{\mathcal {D}},t])}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ae9f6841507f02a5dc078e8f4c6d04a0f962ea)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{0}^{1}e^{x(i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efd3d8beeafd9cb436dc2d9e3685a0a2bb70a69b)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{0}^{1}e^{(ix)\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64523bfe3fec3f20dedc9fad4db794d6d9384538)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{\infty }{\frac {e^{(it)x}{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}}{t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf66423638fcc693390820e78906c6214500951)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\left({\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]\right)\left({\mathcal {F}}_{t}\left[{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\right](x)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/798b053d703484b68dbb0735298996a2fef59f3f)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{\infty }{\frac {e^{(it)x}{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}}{t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8254bc71ab961570d33056a00a76dedeaed14456)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\left({\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]\right)\left({\mathcal {F}}_{t}\left[{\frac {\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}{t}}\right](x)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86799bc67b1304269958995c7f58f697fe239fc2)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }e^{(it)x}{\frac {\partial ^{2}({\text{SurvivalFunction}}[{\mathcal {D}},t]{\text{MeanResidualLifeFunction}}({\mathcal {D}},t))}{\partial t^{2}}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b5f954e3d4c0f01774124be7899188d833d1820)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]={\sqrt {2\pi }}\left({\mathcal {F}}_{t}\left[{\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t^{2}}}\right](x)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8afc6e354ffb180f94158fc72e851f221359a834)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\left(-{\sqrt {2\pi }}x^{2}\right)\left({\mathcal {F}}_{t}[{\text{SurvivalFunction}}[{\mathcal {D}},t]{\text{MeanResidualLifeFunction}}({\mathcal {D}},t)](x)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5522b8de685ff12914e1b2c83073a221188a8a98)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {(ix)^{k}{\text{Moment}}[{\mathcal {D}},k]}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1147968497549393925ccd1fe8601256a46f996)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=e^{(ix){\text{Mean}}[{\mathcal {D}}]}\sum _{k=0}^{\infty }{\frac {(ix)^{k}{\text{CentralMoment}}[{\mathcal {D}},k]}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9097e59def34e345ee0bceb03598c91b9137521)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left(-1+e^{ix}\right)^{k}{\text{FactorialMoment}}[{\mathcal {D}},k]}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bca786c5198e9614a904978bdb30b18b24cec61b)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\sum _{k=0}^{\infty }{\frac {\left(1-e^{-ix}\right)^{k}{\text{AscendingFactorialMoment}}({\mathcal {D}},k)}{k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a89a922c9a0821674602d197eecba789f3f569)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left(\sum _{k=0}^{\infty }{\frac {(ix)^{k}{\text{Cumulant}}[{\mathcal {D}},k]}{k!}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b3d23fcbe01d9ba0255f8f6fe1de1d2f2e491da)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left(\sum _{k=1}^{\infty }{\frac {\left(-1+e^{ix}\right)^{k}{\text{FactorialCumulant}}({\mathcal {D}},k)}{k!}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e361ce3a520dc4894baf11812f4760324f259798)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left(\sum _{k=0}^{\infty }{\frac {\left(1-e^{-ix}\right)^{k}{\text{AscendingFactorialCumulant}}({\mathcal {D}},k)}{k!}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0310376408e49308ebac904f5429f72b97139b)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{MomentGeneratingFunction}}[{\mathcal {D}},ix]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07992b4e669ae29b580c9724c66c721b1c32909)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=e^{(ix){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},ix]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be48c3ae4cbc335d7e2a0d5ce8f18a2aea2b2200)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{FactorialMomentGeneratingFunction}}\left[{\mathcal {D}},e^{ix}\right]{\text{/;}}-\pi <\Re (x)\leq \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/db2585bd7b017b09f4b4875160bb2b8e22c86736)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{DescendingFactorialMomentGeneratingFunction}}\left({\mathcal {D}},-1+e^{ix}\right){\text{/;}}-\pi <\Re (x)\leq \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c43bfb70a7d8ae98d3374092c7224c0192f2500)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]={\text{AscendingFactorialMomentGeneratingFunction}}\left({\mathcal {D}},1-e^{-ix}\right){\text{/;}}-\pi <\Re (x)\leq \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/df8b8e8260d1c723a3e064b8e2a04bdf59cbc597)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},ix])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/932965afde5d7abff5b67afb2ccd851894813078)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left({\text{FactorialCumulantGeneratingFunction}}\left({\mathcal {D}},e^{ix}\right)\right){\text{/;}}-\pi <\Re (x)\leq \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe92e8f5e652398d82a2c972c168ce7b69e71da)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left({\text{DescendingFactorialCumulantGeneratingFunction}}\left({\mathcal {D}},-1+e^{ix}\right)\right){\text{/;}}-\pi <\Re (x)\leq \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c94c89a16951b0c91fe98e0e4fe9a4ff87076af)
![{\displaystyle {\text{CharacteristicFunction}}[{\mathcal {D}},x]=\exp \left({\text{AscendingFactorialCumulantGeneratingFunction}}\left({\mathcal {D}},1-e^{-ix}\right)\right){\text{/;}}-\pi <\Re (x)\leq \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/60416e8ccff318f370b54eb8b3a98ea782276fc1)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b09adde8504f230db7d14b9e8ffd3e233ef1ba2)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=\int _{0}^{\infty }{\text{PDF}}[{\mathcal {D}},t+x]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62294f8d21534948708a8803c312bca83e9a0477)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }\theta (t-x){\text{PDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b94eaedd4fad4fc77fe977c23011c8233ee56b2)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=\int _{-\infty }^{\infty }{\text{Boole}}[t\geq x]{\text{PDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3cf5310bed1f932c52d4b46a122ddc81bd5fe79)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-\int _{-\infty }^{x}{\text{PDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0553de7274a213f3d52c0ed3e55be9ed69ee7adf)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]{\text{PDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5402fc950450219350f779235f2fc83777233dd)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-\int _{-\infty }^{\infty }\theta (x-t){\text{PDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7a7c8bd463d14fe39f520ff6d6d38d0ffdc176)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{CDF}}[{\mathcal {D}},x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60ade7782908f733f672a7fdc614532e74035ed8)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(it)\tau }}}d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c38e8b1a0b4971c4f66f31d8e55d9b45ffa451)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37e8dc4dd83a2d62f03dad1a4f0b7e9ab3ae8d75)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{\infty }\theta (x-t)\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf21c0039f4c2160a3f7ac39239a62bfbe8bd18)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eebeddc10006cfea9ebe65959d36b9c3644be21)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7740b5976ee5a807d28a64b0253fa797586350bc)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=\exp \left(-\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\,dt\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/681aa80b0f1e742d8ec7c84610170dae0b1aabab)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{InverseFunction}}[t,{\text{InverseCDF}}[{\mathcal {D}},t]][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60586ef79387465a3f411d770317537ddf9c8c82)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{InverseCDF}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0be49de3f55909ad6b02f4f792f0f999edd88aba)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{InverseFunction}}[t,{\text{Quantile}}[{\mathcal {D}},t]][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd809c4ceea5c51e315884d915f5d24fde186cb9)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{Quantile}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3816ba60d99237110315d390bc5cea82d38f6d1e)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{InverseFunction}}[t,{\text{InverseSurvivalFunction}}[{\mathcal {D}},1-t]][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b996fe72356ba450c3e8a375be094147fa75222d)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/658d3a561f176890d8a282e2f092d184b67f3ece)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{InverseFunction}}\left[t,{\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\tau )\,d\tau \right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d4562658205f29e364d0f8bfbe87ef2c4cb300c)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau \left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\right)}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/900886fbbfd8208f62fca81c01c6572f482d2437)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aabe82e3841981c48d7c4c7bb01bdb0481f00275)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve1}}({\mathcal {D}},t)}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve1}}({\mathcal {D}},x)}{x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e77ced5f98af2558b2af9a07cd4f6f111e44a9ff)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}{t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5476c582d94cd95af18e1f047f57257dfc6fb75c)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8ccc4e1270a8d70e1ea005c3c18a41035fba5c)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=\exp \left(-\int _{-\infty }^{x}{\frac {1+{\frac {\partial {\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}{\partial t}}}{{\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}}\,dt\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd3c05f24fb0ebf19cb98fd5eb5e94fd97315b5f)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=-{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},x)(1-{\text{CDF}}[{\mathcal {D}},x]))}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28a9745d39dac2a5012ba35d587c2406a6865dfd)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1d62ff732ba0ef04d6d6f74567cb2cc7b69cb2)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{-j+k+1}\right)}{(k+1)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b91edfe39e0e077cd642c5013e7658b6a280e9f)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6501638f2f8fb668b4c25fe3c5a9ec2b25bf81)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a58897eaf47b53b6a4cebfc651fb9205a4ad9a)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b96d9cd03772243f9286f537ece049cf6ef254)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d201c7c571a00d5edc31ec313f104dd0f565ce7f)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a9c4d309056e2394262941c5245997a0fffc23)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},\tau ]}{e^{t\tau }}}d\tau dt}{(2\pi )i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96a2dd3087064049a872e39834d94eb44fe2b808)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3209403aa1587a5364010c0aebf3891e7a5e3d)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91c1139c0647a3ec3e4a43e195bac394942f53a7)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{(i({\text{Mean}}[{\mathcal {D}}]-t))\tau }{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac9b3b138626a1cc061611190c3acabd0424cad)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48d8124b45ffbe5cbd114167337621d62511dc70)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7ef10b6088643a563593564ab85b9fbd4490f03)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ]-(it)\tau }d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0f01e46f9250ed4fbbcb18a40cb923cdc5ea34)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]=1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\,dt}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa4bd445d8670a0ee56930ec73dc56f4ac6ac2f)
![{\displaystyle {\text{SurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/798916af9fb219a9ea6869e8a40705102cbab574)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea2121fa0ea24d7056bc91bbd0544124ae614c5)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-\int _{-\infty }^{x}{\text{PDF}}[{\mathcal {D}},t]\,dt\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70ce4b4f12de629a4db23684e2c36a3b56531476)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]{\text{PDF}}[{\mathcal {D}},t]\,dt\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15ab4f99e6bc3bc503e82fd9e595f18d2989afa)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-\int _{-\infty }^{\infty }\theta (x-t){\text{PDF}}[{\mathcal {D}},t]\,dt\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6da451dc74c75b42d66402ca614fca703c5a45ab)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log(1-{\text{CDF}}[{\mathcal {D}},x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11f2b765c87a953b1520d30b02f0a52b0ee640c)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(it)\tau }}}d\tau dt}{2\pi }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e55c665b4d95aa626495573fb8e703ab5b4c5514)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\,dt}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a46246c1b471e0a2d36c18dcfdeb416569e7eb40)
![{\displaystyle {\text{CDF}}[{\mathcal {D}},x]=-\log \left({\frac {\int _{-\infty }^{\infty }{\text{Boole}}[t\leq x]\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7c067920bc5aee77cf909d062be6133b9e42e2d)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{\infty }\theta (x-t)\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94f73ddce0d1ba9ab135087237f2f35cfa09a1eb)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/864030a79fd97ed26c376185fe4d84ab7307b923)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log({\text{SurvivalFunction}}[{\mathcal {D}},x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffaceff7bd209f219275b25971e35b47858974bc)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35a4ee928b996944ff0553d4818af8ca497a5516)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log(1-{\text{InverseFunction}}[t,{\text{InverseCDF}}[{\mathcal {D}},t]][x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ede840d5ceea8c0e6f39747453a63fcb25e49ea)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{InverseCDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b353fadad7aead144e517ca3fcc1ef5ff42e086e)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log(1-{\text{InverseFunction}}[t,{\text{Quantile}}[{\mathcal {D}},t]][x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1cf358b272b2e45ddc89e54489d3eb9cce7bb17)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4de42e8f5851f7dfdb0df8f19a11311c9a754ab4)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log(1-{\text{InverseFunction}}[t,{\text{InverseSurvivalFunction}}[{\mathcal {D}},1-t]][x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38650a9027dc72d6e2341ab7d147c23d52954857)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\tau (i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])}\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6305b242d2bafd6181a7c98255d30552d6c2b86a)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\text{InverseFunction}}\left[t,{\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\tau )\,d\tau \right][x]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14f95517be8111fe22e01a73299ea1c05f2f51dc)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\right)\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f9bdfa06d360e3e357fabe75526092a8deecd88)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\,dt\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b220c371a79ca21ef177f1fd8bcbaed85441b148)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve1}}({\mathcal {D}},t)}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve1}}({\mathcal {D}},x)}{x}}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a590e8354a7117dfecaaf2f2949968623b1d40e)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}{t}}\,dt\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/918a8c1480f3f9d871cbf383d60eca3e150d20a1)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\text{Mean}}[{\mathcal {D}}]\left(\int _{-\infty }^{x}{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{t^{2}}}\,dt+{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{x}}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cfd7b59b6c26d497a6369bebc8f465405f7c459)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(-{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},x){\text{SurvivalFunction}}[{\mathcal {D}},x])}{\partial x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62642f5fdf578e84bb1c4c4cdbdf6be499b7d272)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/593a4882244565f017b9c54384da7841291b14d8)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{-j+k+1}\right)}{(k+1)!}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4eafd80e33542d7bfe96bd3cd11869da51fdcddf)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/777e8465a8f7cfef934b9a84e0c374bded6237de)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afd6a600357f1aef6dea6a828c535333828aa363)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c826d5ce64ea9cee25ea1232a111be3bc7c46cc)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}(1-{\text{sgn}}(x))-\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9180fa6ff3a7d596bd47d0178e768d32d4f517f)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},\tau ]}{e^{t\tau }}}d\tau dt}{(2\pi )i}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cab536f90a147831ba6b49d5c77d14d98d280faa)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\,dt}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b80d9b4f65554f0a2163c479d7b02839fa30df32)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eff9b39fcc1f64f80c362a8ab52373a80a540aaa)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{(i({\text{Mean}}[{\mathcal {D}}]-t))\tau }{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]d\tau dt}{2\pi }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/204664f5132344598731d2e2ac32676823b44272)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\,dt}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b351fca212b4de4565cab550e107226c8426d863)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a31c320e246be85fc3531c304a353ba92a4337d)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ]-(it)\tau }d\tau dt}{2\pi }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db568afe86d5776b0cd6ea51170863625168dd35)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left(1-{\frac {\int _{-\infty }^{x}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\,dt}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47365f994eb244322fb0d575b62c6ef1f3ebb4b9)
![{\displaystyle {\text{CumulativeHazardFunction}}({\mathcal {D}},x)=-\log \left({\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c918a1f4363141227da2ba9af3ae3c19c046ace)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{PDF}}[{\mathcal {D}},x]}{1-\int _{-\infty }^{x}{\text{PDF}}[{\mathcal {D}},t]\,dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1030b05db24f1b3e92e3f9514467230ce46ad68f)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{PDF}}[{\mathcal {D}},x]}{\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e703c40a82c9efea2a44aef23f52d3d84cae333a)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\frac {\partial {\text{CDF}}[{\mathcal {D}},x]}{\partial x}}{1-{\text{CDF}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/781df1221efce9e556282be69a172f9de0b79306)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]=-{\frac {\partial \log(1-{\text{CDF}}[{\mathcal {D}},x])}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0d6a3df431ad3f58d5a3f281d1e89d86ab0e24)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},t]}{e^{(it)x}}}\,dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a15e363baeea2e458642dff2f66aeab541cbd41)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\left({\mathcal {F}}_{t}[{\text{CharacteristicFunction}}[{\mathcal {D}},t]](-x)\right)}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8d565a70622fd8676a65ece69d8b9c03455a2a8)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]=-{\frac {\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},x]}{\partial x}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac816e103bf46fc811c8fdef29c16cc4de1d672)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]=-{\frac {\partial \log({\text{SurvivalFunction}}[{\mathcal {D}},x])}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df7ae96a79c71dbbc36b418291b36499acc7e8f7)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},x)}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e84b2ca91968419d9b21aad208a69e5f0af43537)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e62a17daea2c706babaf18d0b5b9a2984d6586de)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{InverseCDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80222a3a27e2120ff237913e5722adfd16ac4ae5)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}\delta ({\text{InverseCDF}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{InverseCDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84fc44978223666bfdf61a382215c5e56f4b23ba)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{Quantile}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a94ed18dc4b6ff3e277b00640b86685b6adddeb6)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{Quantile}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d37354310c52fd77ab30e87a887113514891c16f)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}\delta ({\text{Quantile}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{Quantile}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3859c1ad528a36ea31c7bb1d47f06d52bbd2fd6f)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a855374852a72088578d5d039a80190c08b36149)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\int _{0}^{1}{\mathcal {F}}_{t}[1]({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06c2860aaac414faa81cda453316b9c7308c4b6d)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}\delta ({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-x)\,d\tau }{{\frac {1}{2}}-{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{InverseSurvivalFunction}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8fc42a13fc710c14dd0be52191b7149b62a171e)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\right)t\right)d\tau dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b6a59b1bd47ff91b9c4d09555e4a3bd2f0f257a)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\int _{0}^{1}{\mathcal {F}}_{t}[1]\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\,d\tau }{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\right)\tau }\,dt}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e06880021e4bf42b6094ff4933e1182c36dd4809)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{0}^{1}\delta \left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},\xi )\,d\xi -x\right)\,d\tau }{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f18c29b63dfc5953824ade43c66ec936e51e435)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},x)}{\partial x}}}{x\left(1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\,dt\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a04e010fec749e2890dcf2141436baa676ce44)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},x)}{\partial x}}}{x\left(1-{\text{Mean}}[{\mathcal {D}}]\left({\frac {{\text{LorenzCurve1}}({\mathcal {D}},x)}{x}}+\int _{-\infty }^{x}{\frac {{\text{LorenzCurve1}}({\mathcal {D}},t)}{t^{2}}}\,dt\right)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec472d29430752c5b068e9e7afb4bb2254a337ad)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{\partial x}}}{x\left(1-{\text{Mean}}[{\mathcal {D}}]\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{\partial t}}{t}}\,dt\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f39331b6791ea3e3ad17b9a9d8120a6e494242ba)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},x)}{\partial x}}}{x\left(1-{\text{Mean}}[{\mathcal {D}}]\left({\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}{x}}+\int _{-\infty }^{x}{\frac {{\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t])}{t^{2}}}\,dt\right)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/930010183418380e975d25cb4f3fdd0b0f302168)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]=-{\frac {\frac {\partial ^{2}({\text{SurvivalFunction}}[{\mathcal {D}},x]{\text{MeanResidualLifeFunction}}({\mathcal {D}},x))}{\partial x^{2}}}{\frac {\partial ({\text{SurvivalFunction}}[{\mathcal {D}},x]{\text{MeanResidualLifeFunction}}({\mathcal {D}},x))}{\partial x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa181b513aca14406f4ada4d5483c51c7f14dcda)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k]}{k!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e76694ea13b90e40e15b5b8f59d4fffe17e4d70c)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k}{j}}{\text{Mean}}[{\mathcal {D}}]^{k-j}\right)}{k!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef6d5ab19330908e7570abf8fcaa25e4beb80584)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k}{\mathcal {S}}_{k}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{k!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e0273fe14e1be7289804535be11ec9410378095)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{j=0}^{\infty }\left((-1)^{j}{\text{AscendingFactorialMoment}}({\mathcal {D}},j)\right)\sum _{k=j}^{\infty }{\frac {{\mathcal {S}}_{k}^{(j)}\delta ^{(k)}(x)}{k!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abe5d352daaa2c0ba0963cb0704affa3dc2f71e7)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}[k,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k\}]]}{k!}}+\delta (x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/959e280111d009d2f38b84b0cf151f518352d1d2)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k\}\right]\right]}{k!}}+\delta (x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62f3778a9e15be67a177f421d53523a3d3978908)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k\}\right]\right]}{k!}}+\delta (x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cec3d95311addb005787ed143a83db344f479fab)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},t]}{e^{tx}}}\,dt}{((2\pi )i){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a504aed87642046c68d9c2202255d53c786cd722)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},it]}{e^{(it)x}}}\,dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2800a7bcce0dff30f28075de7dfa6459aa852de9)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\left({\mathcal {F}}_{t}[{\text{MomentGeneratingFunction}}[{\mathcal {D}},it]](-x)\right)}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3b99b75ec98758efbe36e28245979bcb6b686fe)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-i\infty }^{i\infty }e^{t({\text{Mean}}[{\mathcal {D}}]-x)}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},t]\,dt}{((2\pi )i){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dac2dcca12407a2855711853601abe0db918048)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }e^{(it)({\text{Mean}}[{\mathcal {D}}]-x)}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},it]\,dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9601646ccd2e3ac8b3fe9f29ed531a6ed8cbbccb)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\left({\mathcal {F}}_{t}\left[e^{(it){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},it]\right](-x)\right)}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7565d556c7ef3b06cff72e4a4ef37360ee4e6fce)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{\infty }e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},it]-(it)x}\,dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e583ad6b96bdaf5ecd917b92e8ba31b1641ecb08)
![{\displaystyle {\text{HazardFunction}}[{\mathcal {D}},x]={\frac {2\left({\mathcal {F}}_{t}\left[e^{{\text{CumulantGeneratingFunction}}[{\mathcal {D}},it]}\right](-x)\right)}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90162296250b2b1c1c377a709dee3ec3d8007209)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\int _{-\infty }^{t}{\text{PDF}}[{\mathcal {D}},\tau ]\,d\tau \right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c02db12b544b98a187f2b06d8e3cb7fe2a11bc5)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{\left(i\left(\int _{-\infty }^{\tau }{\text{PDF}}[{\mathcal {D}},s]\,ds-\xi \right)\right)t}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d92ced50719c2ace980e34e4aae193f82439940)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{CDF}}[{\mathcal {D}},t]][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ebc9945af7ab5945532686b427b7604d1fc24c3)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{CDF}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e21c5be20c81198ba1c3f43b456d91cb44dcc6a)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{CDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42e2aad2e52c146d85bf199681b19e3ac03d925b)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b9d7ad1de0cf46878496fd81c6f80fb9045ad9)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/837335ec348fbd2b76ca7488f36633e8aca23b3f)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }{\text{Boole}}[\xi \leq t]\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4707fa71caa8fff78a790fee8e7e87bea70ddb6)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }\theta (t-\xi )\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd053d8e53d2597b86338347547c562375c3981)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04fd30ccd1febb475c5801c9e419bddc3bba9017)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\xi }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\xi ]}{\xi }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a8103d8b65fced8502fb5a7d2fc436798db6bce)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,1-{\text{SurvivalFunction}}[{\mathcal {D}},t]][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/875d76ed6457d42545bb764795e8afbfdb7363c9)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i(1-{\text{SurvivalFunction}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6366c3bdaea7d743fbaa9a397425ee8ec3d3b8ab)
![{\displaystyle {\text{InverseFunction}}[t,1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},t))][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90e0ffc52db5a59eb9f8e86631eb052c01a3b65d)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{t(i(-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},\tau ))-\xi +1))}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8042d61a59866af66a8053f99d8895ac8ce44889)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,1-\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee704c2e7d5040c21da7f1b62ca9fcc6acbf4b8)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left(-\exp \left(-\int _{-\infty }^{\tau }{\text{HazardFunction}}[{\mathcal {D}},s]\,ds\right)-\xi +1\right)\right)\right)d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/490c9adccbb5c25a05b049de303020ce8661e361)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{Quantile}}[{\mathcal {D}},x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd39abb33a5403770279ba5eee37059aed0abfb)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseSurvivalFunction}}[{\mathcal {D}},1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa4113682624cb4d27e894a5a49abc9fc3a55c0)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{x}{\text{Sparsity}}({\mathcal {D}},t)\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efd113513678e7d60b5414f36fa61db24b9aee45)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\text{LorenzCurve1}}^{(0,1)}({\mathcal {D}},{\text{Quantile}}[{\mathcal {D}},x])}{{\text{PDF}}[{\mathcal {D}},{\text{Quantile}}[{\mathcal {D}},x]]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b792ff4f1fd81e30404366e6b2cf16e58bc534)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},{\text{Quantile}}[{\mathcal {D}},x])}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d49dc1157526b0cf37ea9aacc3cc103e6eceea)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},x)}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5aab1de5c2a234c08dd948a441cf7d11233bb6)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t}}+1\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10349580e078a7afe628ed81faf37bbe86d215b8)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left({\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},\tau ){\text{SurvivalFunction}}[{\mathcal {D}},\tau ])}{\partial \tau }}-\xi +1\right)\right)\right)d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb3d98ccd106b80c36f7f28c3d5c4fd283c7a897)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8ebe41f9cf3382214b4b3c1f64c1cdf4480fc4)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/377f249d2ca73838863aa956dbf1b43f59a1bd6a)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k-j-1}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f152ab3a9cbb80db6d2280940aedc77b68c2974a)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+1-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98518c7339fce8f776920ba080ee4d96d0cb1c1b)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9dbd3de4eb970d84ef38d8144224293d67afb3d)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11883eb2411f7529b03f0138059815277037cdab)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/025d1a964fdba5d93b217ea1e3f79862cbde537d)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25f4cfc3d0832ff1c4b03614bf73be54ceec3421)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0df4790fb52340c97f4e40a05c18e67f99c858)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e51415af36d50c266e188ca4773490effc75a8a)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b513184a1cf6cf72f0c836c987ef9fdce0a933)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8ea8321408c73f754f020060ca88cf6c900f79)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bcddefcf669c761cebe377de8d2169bb1d4074b)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c57c633720cfad7638680ca6f8149f39eb464763)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc1a544a6ea1e9be4bb63d806e0146a666309098)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/166e9848fab2f2bfc3c6d88fffb8e165291726c6)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4bbf8169f8f5e3595c0f7b30b20b69769d61c9)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/521a85f8582bd0cd8fbd782df4c51b81ce8f2e1a)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1af44fd4ed0c40919e3baaada95f0cca5b03a51b)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc62aa3f09b1d5d33fe2d03d1e7280e809f9382c)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb5caed3ab87735cc68cb165ba996aca3f27c48)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf89db69d4fde8f48ca0ceebcfe1e4aca1b601b)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a0f04b3e943242043d87a25763fbdecdfc3d55d)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae903fcc3f9e1a68ce017a5c91c513752324573)
![{\displaystyle {\text{InverseCDF}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1116b779458a0deaa96cafe7830c2600512876a)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\int _{-\infty }^{t}{\text{PDF}}[{\mathcal {D}},\tau ]\,d\tau \right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8fb2191b533d7656267fd03f66babf48970497)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{t\left(i\left(\int _{-\infty }^{\tau }{\text{PDF}}[{\mathcal {D}},s]\,ds-\xi \right)\right)}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13e996e9fdda33f19fb12b039fba806272a38d23)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{CDF}}[{\mathcal {D}},t]][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/953632294811bffc80ea1fe6214ae2f17c0e14e1)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{CDF}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3702d15c14c5a3697309b20e86c7297141fe89d5)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{CDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8aa0d7c150a468fc27d1158f5dcd5ad8be64bc8)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b866a1781f446d81997beebd8482b20efbe056d)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdb932afc6760be3dc5d3ca947e913d72fe1356)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }{\text{Boole}}[\xi \leq t]\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9d0d91c1b50af9f8b22f5f4ad8b0f7601c69e6)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }\theta (t-\xi )\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d5b22d0b81556aabdc422ff8fc8d0aa061c384)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4427cb33fe5e5723b40ee79b6b3352a04f58b7a4)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\xi }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\xi ]}{\xi }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66b4d15869465480d941cb0a61e1fe4dbb0d579c)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,1-{\text{SurvivalFunction}}[{\mathcal {D}},t]][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cb958ecba6ae297ad7c3a38bb262b030fa4c2bf)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i(1-{\text{SurvivalFunction}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/410cf616b714d5303b028df43238ae898809c910)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},t))][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1d93d8839bc6db48f3c2b3ade7d3b8cc9c5cfc)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{t(i(-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},\tau ))-\xi +1))}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4831e3df08b406b8a48b54fb74552ca643cc2b8)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,1-\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8dbcc5a02014cfa309a3db0b1b014ea19cdf939)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left(-\exp \left(-\int _{-\infty }^{\tau }{\text{HazardFunction}}[{\mathcal {D}},s]\,ds\right)-\xi +1\right)\right)\right)d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee563d349932a97c927c6f0f885118a9d75f74ed)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseCDF}}[{\mathcal {D}},x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fffceeb6f253ae936b2acd3d76af126cc50de5a0)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseSurvivalFunction}}[{\mathcal {D}},1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/061432f89be9cedacc6fed136c3928d0adb6570b)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{x}{\text{Sparsity}}({\mathcal {D}},t)\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffa0fd0a7e6f35278f4c912d97f9e6eeaeba2978)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x])}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4439932c369d9532abbf840d67a7a95bfa0850)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\text{LorenzCurve1}}^{(0,1)}({\mathcal {D}},{\text{Quantile}}[{\mathcal {D}},x])}{{\text{PDF}}[{\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x]]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb12e44f8d3350dc46824fc0207802deee2a77c)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},x)}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/137a603cfa0cd2a1caf8c8c4f7edc31a563bd309)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t}}+1\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66a9b85838603fde3411c3b49d8b8d78af858749)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left({\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},\tau ){\text{SurvivalFunction}}[{\mathcal {D}},\tau ])}{\partial \tau }}-\xi +1\right)\right)\right)d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/277cc354bb34638ff97187415026e67b2ceb91fc)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fa1fce3c6891334fc0d8a1d767ab0dca7832ee5)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03dd3ed564dc3a1ad408b9888a99cd1d481380f1)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k-j-1}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73b862b4007b96eab045251420a3ef75daeed071)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+1-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4fefd98769854e13f71fbafe7ab4c79d8e01c4)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5c3162a2947b8af255ffd2cb648aff9a46320d)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1da84a6cd08766d34403fde1966cca17d483d551)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a018007cec803584df60a509d35f99e325358650)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d871c05002f1d740753424323c0bc960e5ca19f)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d10605b70f7b08c129fcc989f5f90cfd88bb35fa)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba62918f79f0662b38f96d87c70ab393439edbb9)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bc7bfac46e4242404c19b5bec0f0e50a1bb8b6)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f07d8ef0f802c8b2e9768ae20d079e7961cc97ab)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7c9c32c9c8981bb14115ecc4deb540a480a242)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f08690b307650cd3fd653bf9f9fe68d9c6bfa1b)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\in t_{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cbf577da1170a5ba7b3ec3c68421e2ba98c08f2)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\in t_{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15a27205a61049ffc4b891a0a69db8f00ed22b21)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8935eecb40062fad1c9f0991f51575b52f439ac)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d819939f91c31b450cd9e5a27b2c8e5722e600f)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\in t_{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8841eb1d7619e24c1193b0f215b8a2c1adc4d210)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\in t_{-\infty }^{t}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3dafdae7b1cf22bf2ce576fc7410b5057aaffad)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f84859b3ac8101389e069ab648fb2aa55f4e21)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/968ac02b525dfd331b44ed1c12afe048e6993c1d)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d24fd600d8294c9add29f9a6766f6669a917cd29)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01cc132317fe146daad0f4605b9baa57ad686cd2)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56af3e0cb3878299017b127b9e33c15a77ca04fb)
![{\displaystyle {\text{Quantile}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](-x)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b7e0946e1b008107708d9ce642e939241b1605)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\int _{-\infty }^{t}{\text{PDF}}[{\mathcal {D}},\tau ]\,d\tau \right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b79978f3f1ab2b0a8d1448262209caa78baf5fb)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{\left(i\left(\int _{-\infty }^{\tau }{\text{PDF}}[{\mathcal {D}},s]\,ds-\xi \right)\right)t}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4f88e6d3e563193099a845386c7d6e7cf5bc97)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{CDF}}[{\mathcal {D}},t]][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8aee55645ae639a52e384ee6ff3ef8c05492bd3c)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{CDF}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43151a79d9e609e7a223d2745e5deb184bb8886d)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}e^{(i{\text{CDF}}[{\mathcal {D}},t])\tau }\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49a8136409db1012059d979a2f6abf48caeb932a)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95964c118bb490d778d0a33be3eb44156a2a0853)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f1dbcf8e25430fa30261f1ba4f62ac7e06156d)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }{\text{Boole}}[\xi \leq t]\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f98a8fc5bbc2c51b308ebeaf35ba070572555247)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{\infty }\theta (t-\xi )\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\right)\,d\xi }{\sqrt {2\pi }}}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca661f3b11f0c34eae972717ce943ff0c217fdcf)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4de956987ad239deffab242f2c94d568ea6e7097)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\xi }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\xi ]}{\xi }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d367c9e5d80c78fc59fd81b26c56aec77bf848c)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,{\text{SurvivalFunction}}[{\mathcal {D}},t]][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91f7d69b8979f36d67510ae66b8017db7bae7b48)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i(1-{\text{SurvivalFunction}}[{\mathcal {D}},\tau ]-\xi ))t}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46e97c53b95de60a1aec4deaa606989d6fdb5e5d)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}[t,1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},t))][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32b763803f7d2472f6b2ab7dd21611a2659f6914)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i(1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},\tau ))-\xi ))t}d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c198f1e0c58f446d206a923846cefbebc2c8c66a)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,1-\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eeba387544cbf689a10ad8c7cacddc1c0b8b4d64)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left(i\left(1-\exp \left(-\int _{-\infty }^{\tau }{\text{HazardFunction}}[{\mathcal {D}},s]\,ds\right)-\xi \right)\right)t\right)d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6308e92fdfdb1725a153f528e0af39ef5f591295)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseCDF}}[{\mathcal {D}},1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f40d724859453a3329d4cfb8fbaf14e78e8d702b)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{Quantile}}[{\mathcal {D}},1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fcba2d912907df82095799ced3d7551fa8d1d33)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{1-x}{\text{Sparsity}}({\mathcal {D}},t)\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08a12d7cc360657e3ea75a46b6808cf39f39f76a)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]=-{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},1-x])}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/993329a8786d62fd553cda7382a232536ca780c9)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {{\text{Mean}}[{\mathcal {D}}]{\text{LorenzCurve1}}^{(0,1)}({\mathcal {D}},{\text{Quantile}}[{\mathcal {D}},1-x])}{{\text{PDF}}[{\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},1-x]]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a25b881b88e2f4474647910fc92e9ac977646776)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]=-{\text{Mean}}[{\mathcal {D}}]{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},1-x)}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa251d1a30f1740a4826db22a2397afad33d5dc)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t}}+1\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5263a237cb8dfe4929ceb023d1e42cffad1fc53)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {\int _{-\infty }^{1-x}\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left({\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},\tau ){\text{SurvivalFunction}}[{\mathcal {D}},\tau ])}{\partial \tau }}-\xi +1\right)\right)\right)d\tau dtd\xi }{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed3bce3b48ce999167ce6d80b0d4e07488c52945)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/556fd74d1983a7f87e8eadf6414c4d9792589e2c)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}\right)i\right)\tau \right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31d23e7113b6a5773ae62e57481610fd43602a73)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k-j-1}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8622d1c7ae1aefe6ffa45cc271d24fb980a51d)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{-j+k+1}\right)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c21c457f993152797a3f8e45422dfda40df7d4a)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4799adb3e894bdf949e437d643acdeb4680ef5)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8124a834a388bc455159a72ceadebfa23b7f5ddf)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80dc5be72c7b0a674e61fcf1a0dbcbafef5df2b2)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e71cf9ef4d0ef5885ffaa7abc17daac113bbf0)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6433e834db23121af8f8a69331a8589192525894)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e081e4abbd342654f38b45a0420e3bbad8ce6e95)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fecece6dbd5caee4a690cb5fb2954433b58bce0e)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j+1}{\mathcal {S}}_{j+1}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3da55275e500e07a43c64724fc1c76288fe2688a)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d55610ed38ced141b80a869be0258f2ba36d573)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left(\tau \left(i\left(\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right)\right)\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd5ecb19874f5b2c3f5cbbbb874e0c301c2b670a)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abee691d4106ecc259b13a022fc406575840f07d)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2b8895e2c4cda97dffc73b9d4a1e1380567e4e)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de1aa337249592217942512e6343ccce441d3dac)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4471e5a3230439303cc6e23daf11a16dc3c90b6)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0ade0e8f46db8c46913b8555e588a7922cd755)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/214bc7984bdd81153ebfd487c82a2fc04f0e268a)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb60390e761b0affff1309c2f4ca593ab5743fb5)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d099f3d44b1bbf17ff64d9f797dee7699ab555d)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/914d2417d71dee8eace2ff549403cdf1ec752ead)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90cf226680667cde29d8a8137368b0aea78efb90)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][1-x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77411a8e20b1f4dcbd0bf756b4456bbde2279a0f)
![{\displaystyle {\text{InverseSurvivalFunction}}[{\mathcal {D}},x]={\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt}{\tau }}\right](x-1)\right)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f88c67de1193a47bc5fc6d1e9cf495ecb82fee1b)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,\int _{-\infty }^{t}{\text{PDF}}[{\mathcal {D}},\tau ]\,d\tau \right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5d92e336584bf0238811157b263bad1fc274353)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{\left(i\left(\int _{-\infty }^{\tau }{\text{PDF}}[{\mathcal {D}},s]\,ds-x\right)\right)t}d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66312d98ecb94cf3c57a6c0fad85c279c2f70611)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}[t,{\text{CDF}}[{\mathcal {D}},t]][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f40ed75f3999a90a5f531e7e2e1d6d77b9fae670)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{(i({\text{CDF}}[{\mathcal {D}},\tau ]-x))t}d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b4c40db6c18c1782a042a3f1827e626d2e378c)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{t}\left[\int _{0}^{1}e^{t(i{\text{CDF}}[{\mathcal {D}},\tau ])}\,d\tau \right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f910ee71b0cab59c3908c8fcf5035c2c50b3bd6d)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\xi )\tau }}}d\tau d\xi }{2\pi }}\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6faedfc5be76c09db6919529efd736cfc2a063b4)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d62bad38a9a016be0ecac3fe482be44b91f0d0b1)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ae55f4222bb06eb2c468199b1ce87fe8ce64ea)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left({\frac {i\int _{-\infty }^{\tau }\int _{-\infty }^{\infty }{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},s]}{e^{(iu)s}}}dsdu}{2\pi }}-ix\right)t\right)d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07cbbed0ceb1d4fb48854970434841312ef72062)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}+{\frac {i\tau }{2}}\right)\,dt\right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76adf2d9788fb0f30c986d7500183c1216e226cd)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}[t,1-{\text{SurvivalFunction}}[{\mathcal {D}},t]][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3532da2cab08bf0b7c69f1741ab4b265ef3e08)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}e^{t(i(-{\text{SurvivalFunction}}[{\mathcal {D}},\tau ]-x+1))}d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4febb18e28803b9347990c74762059fe9115b2)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}[t,1-\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},t))][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9257cb4f127d9e8466800016cba399789d4c1827)
![{\displaystyle {\frac {\partial {\text{InverseFunction}}\left[t,1-\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b20934ca4e6ef94125521ca0041349311f615fbd)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left(-\exp \left(-\int _{-\infty }^{\tau }{\text{HazardFunction}}[{\mathcal {D}},s]\,ds\right)-x+1\right)\right)\right)d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0676a5d21ced299c70a7cab2eceb5ed3378b6c4e)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseCDF}}[{\mathcal {D}},x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f36b1301a5101d99f66ee6dbce64c235817913cb)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{Quantile}}[{\mathcal {D}},x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da9bdfb8401bfc2c56985ecfe1372aca885bb16e)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseSurvivalFunction}}[{\mathcal {D}},1-x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89bc0f9070f2939a48e7e50d1923208b9050b25f)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\text{Mean}}[{\mathcal {D}}]{\frac {\partial ^{2}{\text{LorenzCurve1}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x])}{\partial x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0cf6606e049c205a9653ac8b85a8756d2744e2)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\text{Mean}}[{\mathcal {D}}]{\frac {\partial ^{2}{\text{LorenzCurve2}}({\mathcal {D}},x)}{\partial x^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/260b8ffb1b9588605a81d8de52f96e67e059f437)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t}}+1\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55da8bfb75c5ab7a0da7ac2bd0e6006992dcd4c0)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(t\left(i\left({\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},\tau ){\text{SurvivalFunction}}[{\mathcal {D}},\tau ])}{\partial \tau }}-x+1\right)\right)\right)d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f35921ee047ad16baf9384b1e8c7829ae60730fc)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}+{\frac {1}{2}}({\text{sgn}}(t)+1)\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bce822afb42234a3a9ce7d036a83d2079ec8173)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{Moment}}[{\mathcal {D}},k+1]}{(k+1)!}}\right)i\right)\tau \right)\,dt\right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51d72c2d9913f67ffe7d3703edec531d06f1d09f)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/453f759f19f331c3ae12cf0fbe4b0f213cef6ea2)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}\right)i\right)\tau \right)\,dt\right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cec0fc846979b990c87d868d6f2be94cc6bacae)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right)\sum _{j=0}^{k+1}\left((-1)^{k-j-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44a9d48053c9ce276070dbcb2295691bd3b0c09b)
}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/576f82b2ece27cd70ddcf49730587015aad63f09)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c381fe96df33646539d411513b3fd77450e9a7be)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{(k+1)!}}\right)i\right)\tau \right)\,dt\right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee16c45bb9d6a7bab004c7909c05fbf9ab5f19a)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e69099f1e1376c265e7db5f0383c10033a57b6fb)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right)i\right)\tau \right)\,dt\right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b98909e38278078047e5c23377723c221fa13ff)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37ef3a3138296b983892a05a3db11fcf6af1ae14)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left(\left(\left({\frac {1}{2}}(1+{\text{sgn}}(t))+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(t)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{(k+1)!}}\right)i\right)\tau \right)\,dt\right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68a2b4e60d9215f370c856fb257df343379f989a)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},\tau ]}{e^{\xi \tau }}}d\tau d\xi }{(2\pi )i}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca7380b55511a1276edf1c3e9ac8ae1f118c35cc)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3182dc3566f698ef84585bb9c3a519c7ab5f765a)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial }{\partial x}}{\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dd5430bbce01ee7e3306b23cc2afab23d487c2c)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left(i\left({\frac {\int _{-\infty }^{\tau }\int _{-i\infty }^{i\infty }{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},s]}{e^{us}}}dsdu}{(2\pi )i}}-x\right)\right)t\right)d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd362e869a8a574e175c096fcbea4ba5177b5e89)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt\right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1da6734cb5db95e56ba59c3cb75c4653ab29b5)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-i\infty }^{i\infty }{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},\tau ]e^{\tau {\text{Mean}}[{\mathcal {D}}]-\xi \tau }d\tau d\xi }{(2\pi )i}}\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e1ad6f2770a317f5b58203fc1d52429dc5df7b)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bad94f152f7c285e001d512d57a468bf82501d3c)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0125c2a0f6011a02e733fdecb082418f82841d68)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left(i\left({\frac {\int _{-\infty }^{\tau }\int _{-i\infty }^{i\infty }e^{s{\text{Mean}}[{\mathcal {D}}]-us}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},s]dsdu}{(2\pi )i}}-x\right)\right)t\right)d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9695ccdf3df60bd52fc5b35b76d0eed6eebb86bd)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt\right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc80690a1dea0ecc00ad0abe11b3a23072a677c)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}\int _{-i\infty }^{i\infty }{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},\tau ])}{e^{\xi \tau }}}d\tau d\xi }{(2\pi )i}}\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9dc83efd00ae7e5fd142df8b79c466229edf80)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {\int _{-\infty }^{t}{\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},\tau ])](-\xi )\,d\xi }{\sqrt {2\pi }}}\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb2d5bf656f2b18ef40d1a5cd021946769da4fd)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\partial {\text{InverseFunction}}\left[t,{\frac {1}{2}}+{\frac {i\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right][x]}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2851590bf73db678e365d80a7c7a58371817fb4)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{0}^{1}\exp \left(\left({\frac {\int _{-\infty }^{\tau }\int _{-i\infty }^{i\infty }{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},s])}{e^{us}}}dsdu}{2\pi }}-ix\right)t\right)d\tau dt}{2\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6023cfba918f8cb589ed107756f23a925bf4f4a)
![{\displaystyle {\text{Sparsity}}({\mathcal {D}},x)={\frac {{\mathcal {F}}_{\tau }\left[\int _{0}^{1}\exp \left({\frac {i\tau }{2}}-{\frac {\tau \left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-t)\right)}{\sqrt {2\pi }}}\right)\,dt\right](-x)}{\sqrt {2\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/004e7fe6e8e9ec23b540693bb631cb9fa6b79bb1)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{-\infty }^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/769986c4c743d262c1d2a90bc1ff88bdf38bba31)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bacef4a93fc3e974ae24b1422ddcaf7d080a0d88)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)=\left({\begin{array}{cc}\{&{\begin{array}{cc}0&x<u\\{\frac {\int _{u}^{x}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}&u\leq x<v\\1&{\text{True}}\\\end{array}}\\\end{array}}\right){\text{/;}}u={\text{support}}[{\mathcal {D}}][[1,1]]\land v={\text{support}}[{\mathcal {D}}][[1,2]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65c98155acdff5bb2811ea3a653886aede2cf512)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d09c98890c13094b0ad291ca1e50b3aafe6c961b)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {x{\text{CDF}}[{\mathcal {D}},x]-\int _{-\infty }^{x}{\text{CDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35e0034f4d8e8ee1db0615f00938221155a55595)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{x}{\frac {t{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\tau )t}}}dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb724bee5c02ab531518b6b2aa0be8191338412)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}t\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4d3f2a37c5e26baee8408e0765a5fb206d7820)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {(1+\tau (ix)){\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau ^{2}}}\right](-x)}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ad04192d92ea2e3cad2c4093bfc53f65b55266)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)=-{\frac {\int _{-\infty }^{x}t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b86da724b5e0dbd53c4f3435f2938ae37c1f7f72)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {x(1-{\text{SurvivalFunction}}[{\mathcal {D}},x])-\int _{-\infty }^{x}(1-{\text{SurvivalFunction}}[{\mathcal {D}},t])\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9862a917f9e59126e4f319852236f26f3216dc6d)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84d5876b6c094c9d00dafaa4d5874441a2a4a0ab)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\left(t\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right)\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd9da45db9e01110b3b55e6baf883f47302f0e34)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{InverseCDF}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b1b2cfe528f88d80c95ce0ea57362fd9c638419)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb9b2bddc4dfe709f74c54288e308b17f2ee4be)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{Quantile}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adec9e61c48f13a0b7ea2005a6ec7ef728c48817)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}{\text{Quantile}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ceb822655048597611fc90bce20d86683ff7bc)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e9ad5083373b9a39dcb46ef04cd45652983061)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{{\text{SurvivalFunction}}[{\mathcal {D}},x]}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48f582af74fea943bde75db9fe35a7ef0cf36af3)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}t\exp \left(\xi \left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},s)\,ds-t\right)\right)\right)d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4709ab6a41012be977e14ce938f1f94e2d4a7f94)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed6cc573808a9272c77e280e79da58ecab547179)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee0e28b2397e1d5214734412099c9452c7cdc176)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)=1-{\frac {({\text{MeanResidualLifeFunction}}({\mathcal {D}},x)+x){\text{SurvivalFunction}}[{\mathcal {D}},x]}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbfe4e7a205515a93a6dba377e22188d5ca2ce69)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right){\text{Moment}}[{\mathcal {D}},k+1]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31affed18bafb323b2e040cce921b5288c4c2f18)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{n-j}\right)}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a5b33a8cb8b10235b8f821675f71c37ca4df3cf)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9aa96c5e5fe93edfd20ef23b550e3620d13be3b)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right)\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/593a7f6dc0b745bd53f26fb5176422b80e7c0e99)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c63956c363f1291eebd0cca604d1c1e89be75a)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83dd7e97e96bae4a877a268b9fd21d66b025594c)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}(x)\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e19e867d58c3333c24fac882e65fcbb530d4a4f)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{x}{\frac {t{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\tau )t}}}dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4fe186c7332eede10baf05336b6973e92a8eaa)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {(1+\tau (ix)){\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau ^{2}}}\right](-x)}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01db2dce4cca21cd61751388ddfad905c2e2ad35)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{x}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\left(te^{(i\tau )({\text{Mean}}[{\mathcal {D}}]-t)}\right)dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04f299ea31f5a219253fc6bf879e5546a0c44893)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\left((1+\tau (ix))e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}\right)}{\tau ^{2}}}\right](-x)}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fddc09117ee33ded55e42d360cc3b45b2a68f6d0)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{x}t\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ]-(i\tau )t)dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75ee95845cf7bc886f5df553e7a44c336e17c87d)
![{\displaystyle {\text{LorenzCurve1}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {(1+\tau (ix))\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau ^{2}}}\right](-x)}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7c24d1584d2d3a3457bbed597effaa374d79b3)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{-\infty }^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/574f56479e5a3c7bb48c7ebc2a78e1b1f9eb1ab2)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e502449cebf0c594291413cf0168ffce0f601cc5)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c9b6c1866aa9dff108094f6d45a6fc4498579d9)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {x{\text{InverseCDF}}[{\mathcal {D}},x]-\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\text{CDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e7d10316dd2234fdd3ef82de03a2b1270502607)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\frac {t{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{e^{(i\tau )t}}}dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ff76606cb61c01417202da3f2236e1e67aee8b2)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e0bcb7438a9b5f8fb89700bb75c3316abaccb8)
)}{\tau ^{2}}}\right](-{\text{InverseCDF}}[{\mathcal {D}},x])}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86080846ff9a42e8b719a78273b98c7e9e06d412)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)=-{\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a02bc630e902dfecb1618bf4989058039c730520)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {x{\text{InverseCDF}}[{\mathcal {D}},x]-\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}(1-{\text{SurvivalFunction}}[{\mathcal {D}},t])\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11e304bea76df396bc2bcb1e0b28ee624945eb0c)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b994dc66cd8922d247415cff252ec255b1ff167)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\text{HazardFunction}}[{\mathcal {D}},t]\left(t\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right)\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de661abb073fc8362c21c0936ae4a2bcab9ae556)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{InverseCDF}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c3b4c231660735680f4d5a721491ec84f3871c4)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{0}^{x}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7c96dd38c189aa5cef589db7d8b37840f82fcb)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{Quantile}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/658b7dfaa1f72d536885f545134ed7c7fa19bfcf)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{0}^{x}{\text{Quantile}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3056cdcb38c1288bba36515392ea6758ebedbbb5)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}\int _{-\infty }^{\infty }\int _{0}^{1}te^{\xi (i({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-t))}d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3b58efe0302001e663b422822cba6a3cada011)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{1-x}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25bb4ee5a9f485598d870cc47809f89cc94ae364)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}\int _{-\infty }^{\infty }\int _{0}^{1}t\exp \left(\xi \left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},s)\,ds-t\right)\right)\right)d\tau d\xi dt}{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe3cf1a0633f2ad28adb1baa2649aa77e5d6b0f)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{0}^{x}\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec216a94b0596a07cab4bf66742dd2124c324a55)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\text{LorenzCurve1}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ca9969bbfba875193d39669b3721bd71c82e51)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t{\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t^{2}}}\,dt}{{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5e152a5b9d83355dc1894967dbbdf0c10c9984)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {{\frac {{\text{InverseCDF}}[{\mathcal {D}},x]{\frac {\partial ({\text{MeanResidualLifeFunction}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x]){\text{SurvivalFunction}}[{\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x]])}{\partial x^{1}}}}{\frac {\partial {\text{InverseCDF}}[{\mathcal {D}},x]}{\partial x}}}-{\text{MeanResidualLifeFunction}}({\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x]){\text{SurvivalFunction}}[{\mathcal {D}},{\text{InverseCDF}}[{\mathcal {D}},x]]}{{\text{Mean}}[{\mathcal {D}}]}}+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71f1ee049a040d49aebe640df6bf41f63543a211)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {{\text{Moment}}[{\mathcal {D}},k+1]\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right)}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/972312056cfe2a8019987bc4f187f652dcd71515)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right)\sum _{j=0}^{k+1}{\text{CentralMoment}}[{\mathcal {D}},j]\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{n-j}\right)}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba91d77fc2378ebefcfe9cc4bd323acb1d8fd76)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right)\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7acf0443378dca0667bf2b0000d20f361d1e0ce3)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\sum _{j=0}^{k+1}\left((-1)^{-j+k-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right)}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c28f4f36a9a722eed730d06aa8a035eb57fd5149)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right){\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95e5688183c9c6f10afb16a530ac03c541c777a3)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta ({\text{InverseCDF}}[{\mathcal {D}},x])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54ece97bb2cc33693ea517cd7fadd25385709595)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\sum _{k=1}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k-1)}({\text{InverseCDF}}[{\mathcal {D}},x])\right){\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]}{k!}}}{{\text{Mean}}[{\mathcal {D}}]}}+\theta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3957d8ce167fe508489c3adbb904fbba0e5aae5a)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\frac {t{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{e^{(i\tau )t}}}dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47a53dd587e2695ffbd9231b93a3bdac90773215)
)}{\tau ^{2}}}\right](-{\text{InverseCDF}}[{\mathcal {D}},x])}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f72a0f609333cc8267569a89d0a951587f3088e8)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\left(te^{(i\tau )({\text{Mean}}[{\mathcal {D}}]-t)}\right)dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5617fb4e7d3431e638cb0ffe1085a1e743872592)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\left(e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}(1+\tau (i{\text{InverseCDF}}[{\mathcal {D}},x]))\right)}{\tau ^{2}}}\right](-{\text{InverseCDF}}[{\mathcal {D}},x])}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18e9c1e67be2ac4ac0f38f7190fc37da6a0e6c79)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{{\text{InverseCDF}}[{\mathcal {D}},x]}t\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ]-(i\tau )t)dtd\tau }{(2\pi ){\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a804396d17ab4df0f64e17a50a011f9680ddc89b)
![{\displaystyle {\text{LorenzCurve2}}({\mathcal {D}},x)={\frac {1}{2}}+{\frac {{\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])(1+\tau (i{\text{InverseCDF}}[{\mathcal {D}},x]))}{\tau ^{2}}}\right](-{\text{InverseCDF}}[{\mathcal {D}},x])}{{\sqrt {2\pi }}{\text{Mean}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5addcb55af6062e74a677b653c2da2c09786ab4f)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{x}^{\infty }(t-x){\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37c262797983a25ec8913e00f2e01b2d082c3f66)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{0}^{\infty }t{\text{PDF}}[{\mathcal {D}},x+t]\,dt}{\int _{0}^{\infty }{\text{PDF}}[{\mathcal {D}},x+t]\,dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/820b8ecf1dc1126ee32ab9563f5087d24ba3526f)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4be9db1f59bcd4627fdf6b5991cd7ef1e8ef70f)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x-\int _{-\infty }^{x}(t-x){\text{PDF}}[{\mathcal {D}},t]\,dt}{1-\int _{-\infty }^{x}{\text{PDF}}[{\mathcal {D}},t]\,dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/994938c628c928f8b9b363fc5508a6175741a3bc)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=\left({\begin{array}{cc}\{&{\begin{array}{cc}{\text{Mean}}[{\mathcal {D}}]-x&x<u\\-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{u}^{x}t{\text{PDF}}[{\mathcal {D}},t]\,dt}{\int _{x}^{\infty }{\text{PDF}}[{\mathcal {D}},t]\,dt}}&u\leq x<v\\{\text{Indeterminate}}&{\text{True}}\\\end{array}}\\\end{array}}\right){\text{/;}}u={\text{support}}[{\mathcal {D}}][[1,1]]\land v={\text{support}}[{\mathcal {D}}][[1,2]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd1325edceec67516e9d922523eb15655ca7993)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x+\int _{-\infty }^{x}{\text{CDF}}[{\mathcal {D}},t]\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0547f5f91b314efe0abdd184ed6d98d03215c8)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{x}^{\infty }(t-x){\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29d017c6666216cb07415daf38ef318db8d17cbc)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{0}^{\infty }t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t+x]}{\partial t}}\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ced99f2a36a224b599c9e1b8307160b1b14afbde)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}{1-{\text{CDF}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5764296a8762f9e8c27ae5027f45a2c6c37233ae)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad71da78f957df278a9b7837673043ffd3d1cc03)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{CharacteristicFunction}}[{\mathcal {D}},\tau ]}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5427b14de5d9c585787f4bea96738d00c8018c)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x+\int _{-\infty }^{x}(1-{\text{SurvivalFunction}}[{\mathcal {D}},t])\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e1a1ae9fbe28ce01d28781eebe8c69e17683e0)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-{\frac {\int _{x}^{\infty }(t-x){\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/798c43e29b8ed61d680608edad857993a2874130)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-{\frac {\int _{0}^{\infty }t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t+x]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83e0ff7ac8726e5d7c4a1cf9d74777e7fa850d19)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x-{\frac {\int _{x}^{\infty }t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b7b2f8c36b5c69a5589da82cdc7517f9ebcff1a)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]+\int _{-\infty }^{x}t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3189652dc25d59e3670452e8fff9a991b0db4f1e)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-{\frac {\int _{0}^{\infty }{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t+x)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t+x)}}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36b62acf4b95319daf69f89b1ec5e933a1355e48)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x\exp(-{\text{CumulativeHazardFunction}}({\mathcal {D}},x))+{\text{Mean}}[{\mathcal {D}}]-\int _{-\infty }^{x}{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc2a3225f435e44aad23eae0972f5dece3bcdab3)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{0}^{\infty }\left(t\exp \left(-\int _{-\infty }^{t+x}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right){\text{HazardFunction}}[{\mathcal {D}},t+x]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76050454023a5b82047f985004c1ca615ef4c123)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x\exp \left(-\int _{-\infty }^{x}{\text{HazardFunction}}[{\mathcal {D}},t]\,dt\right)+{\text{Mean}}[{\mathcal {D}}]-\int _{-\infty }^{x}\left(t\exp \left(-\int _{-\infty }^{t}{\text{HazardFunction}}[{\mathcal {D}},\tau ]\,d\tau \right)\right){\text{HazardFunction}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f8f11b59a0d11be09acb7066fb1dd30c9ce72e)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{InverseCDF}}[{\mathcal {D}},\tau ]-t-x))\xi }d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/771d07fc958ab90b26a05c503d8c1d78cdf12d26)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4136a687f30c76d32ea66479069787d0e15ac97b)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{Quantile}}[{\mathcal {D}},\tau ]-t))\xi }d\tau d\xi dt}{2\pi }}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da186bf502472f17ae5d12d9a84e66e9f0c50de7)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{Quantile}}[{\mathcal {D}},\tau ]-t-x))\xi }d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/791a63fff27517712e2b2571a66075d7330d8cf0)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}{\text{Quantile}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2372b493dd73edf9f3c35f13c77dbcc0fdf646b)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}te^{(i({\text{InverseSurvivalFunction}}[{\mathcal {D}},\tau ]-t-x))\xi }d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f039b07bf82857c5f7a20b6387682550a823d12b)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {{\text{Mean}}[{\mathcal {D}}]-\int _{{\text{SurvivalFunction}}[{\mathcal {D}},x]}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/beee7c988fb48d5f4ad2e516960c6a1997b7fc48)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{0}^{1}t\exp \left(\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},s)\,ds-t-x\right)\right)\xi \right)d\tau d\xi dt}{(2\pi ){\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9064b98bf27e9f7c76ce5b3ecf658242f2d97752)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x(1-{\text{CDF}}[{\mathcal {D}},x])+{\text{Mean}}[{\mathcal {D}}]-{\frac {\int _{-\infty }^{x}\int _{-\infty }^{\infty }\int _{0}^{1}t\exp \left(\left(i\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{\tau }{\text{Sparsity}}({\mathcal {D}},s)\,ds-t\right)\right)\xi \right)d\tau d\xi dt}{2\pi }}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2aef073fa7900134b53daeb5ff3d2ebbac48a2)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {-x(1-{\text{CDF}}[{\mathcal {D}},x])+{\text{Mean}}[{\mathcal {D}}]-\int _{0}^{{\text{CDF}}[{\mathcal {D}},x]}\left({\text{Median}}[{\mathcal {D}}]+\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},s)\,ds\right)\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bda871153f3e347bffac8d9766e5853c08cc5311)
)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1274b5f598f7d08b5298817a20ecdb9f2cfdcf30)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]\int _{0}^{\infty }{\frac {t{\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t+x)}{\partial t}}}{t+x}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a326d7d7aa257f16fb748b28914282b72c289f16)
)-x+(x{\text{Mean}}[{\mathcal {D}}])\int _{-\infty }^{x}{\frac {\frac {\partial {\text{LorenzCurve1}}({\mathcal {D}},t)}{\partial t}}{t}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7505ab4eccd7eaf98fcd56e7b33c912724646e1)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]-x+({\text{Mean}}[{\mathcal {D}}]x)\int _{-\infty }^{x}{\frac {{\text{LorenzCurve1}}({\mathcal {D}},t)}{t^{2}}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0511ef4de6be415c3589981b5f29d802446e58)
)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5098ea119b72aa96ba97df0665265f71bc78b532)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\text{Mean}}[{\mathcal {D}}]\int _{0}^{\infty }{\frac {t{\frac {\partial {\text{LorenzCurve2}}({\mathcal {D}},{\text{CDF}}[{\mathcal {D}},t+x])}{\partial t}}}{t+x}}\,dt}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ce2b6cdbdecc0fa7ae41180ee2cbae20bd1e187)
)-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)(x{\text{Moment}}[{\mathcal {D}},k+1]+{\text{Moment}}[{\mathcal {D}},k+2])}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b3654ef360747185d9ea1e12e5f7c618dc8bc7f)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{Moment}}[{\mathcal {D}},k+2]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da1a539114c522cade76b608b2b6e5e22fcc3fff)
)-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x\sum _{j=0}^{k+1}\left({\binom {k+1}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+1-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]+\sum _{j=0}^{k+2}\left({\binom {k+2}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+2-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52b2f9ae9289c1d5d58fd223eed5ae1acefd7245)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+2}\left({\binom {k+2}{j}}{\text{Mean}}[{\mathcal {D}}]^{k+2-j}\right){\text{CentralMoment}}[{\mathcal {D}},j]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69cd3ceb7ab1c484aa86e132be854b20a55e09df)
)-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x\sum _{j=0}^{k+1}{\mathcal {S}}_{k+1}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]+\sum _{j=0}^{k+2}{\mathcal {S}}_{k+2}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fca3a438e5ea863269185cc50785a271bf5d9ed9)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+2}{\mathcal {S}}_{k+2}^{(j)}{\text{FactorialMoment}}[{\mathcal {D}},j]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a114205593d7edacf4d183cce5787466645ac0f3)
)-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x\sum _{j=0}^{k+1}\left((-1)^{k-j-1}{\mathcal {S}}_{k+1}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)+\sum _{j=0}^{k+2}\left((-1)^{k-j}{\mathcal {S}}_{k+2}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef1777bda8e63f21af982a88cdc1fd4358ba6d5)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right)\sum _{j=0}^{k+2}\left((-1)^{k-j}{\mathcal {S}}_{k+2}^{(j)}\right){\text{AscendingFactorialMoment}}({\mathcal {D}},j)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bed12f21b4e082722aad5a1d59a8a9a3182e525a)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)({\text{BellY}}[k+2,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+2\}]]+x{\text{BellY}}[k+1,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+1\}]])}{(k+1)!}}+(1-\theta (x)){\text{Mean}}[{\mathcal {D}}]-x\theta (-x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8ca08f06d97c56cde37715ceece7866bbdd3e3)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}[k+2,1,{\text{Table}}[\{1,{\text{Cumulant}}[{\mathcal {D}},j]\},\{j,k+2\}]]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dac911a53960d7d9b85e8699ac7b06ff0fb42d6)
)-x\theta (-x)+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left(x{\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]+{\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]\right)}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/680cddfde9b2b42d800fb020fd4bf79263105a98)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=1}^{j}{\mathcal {S}}_{j}^{(i)}{\text{FactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/062b60b58785bbf9bb86b22a77fa2d2e3389923c)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {\sum _{k=0}^{\infty }{\frac {\left((-1)^{k-1}\delta ^{(k)}(x)\right)\left({\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]+x{\text{BellY}}\left[k+1,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+1\}\right]\right]\right)}{(k+1)!}}+(1-\theta (x)){\text{Mean}}[{\mathcal {D}}]-x\theta (-x)}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d460146c46d4406157c2e63f330dff10f3ea54)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\theta (-x){\text{Mean}}[{\mathcal {D}}]+\sum _{k=0}^{\infty }{\frac {\left((-1)^{k}\delta ^{(k)}(x)\right){\text{BellY}}\left[k+2,1,{\text{Table}}\left[\left\{1,\sum _{i=0}^{j}\left((-1)^{j-i}{\mathcal {S}}_{j}^{(i)}\right){\text{AscendingFactorialCumulant}}({\mathcal {D}},i)\right\},\{j,k+2\}\right]\right]}{(k+1)!}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9e053a35158f961ff52980004ce42d6aafeed44)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\frac {1}{2}}({\text{Mean}}[{\mathcal {D}}]-x)-{\frac {{\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau ^{2}}}\right](-x)}{\sqrt {2\pi }}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed6f6945b18a227ca31f078d08fc01ca71cf703)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1dcde32817a42d85c01339dee9e611e775a2c9c1)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {{\text{MomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0b4836a76e4a3b66ebe2d949e31c3df06b8216)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\frac {1}{2}}({\text{Mean}}[{\mathcal {D}}]-x)-{\frac {{\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau ^{2}}}\right](-x)}{\sqrt {2\pi }}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e834d95028aa6fad56a6f5b26fc2c037d77310c8)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8076103c78ba50676aa9f5db33779f0d6a07cecf)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }\left[e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]\right](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {e^{(i\tau ){\text{Mean}}[{\mathcal {D}}]}{\text{CentralMomentGeneratingFunction}}[{\mathcal {D}},i\tau ]}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5effa02e08be9a9bd8fd129c5787dd3dcf0d0d)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)={\frac {{\frac {1}{2}}({\text{Mean}}[{\mathcal {D}}]-x)-{\frac {{\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau ^{2}}}\right](-x)}{\sqrt {2\pi }}}}{{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9dfeee82b7372907e5ec13ad30fb0b0db4b631b)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\right)\,dt}{{\sqrt {2\pi }}{\text{SurvivalFunction}}[{\mathcal {D}},x]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51589b664ec465abe311bec2a396237cc16c2e58)
![{\displaystyle {\text{MeanResidualLifeFunction}}({\mathcal {D}},x)=-x+{\frac {2\int _{x}^{\infty }t\left({\mathcal {F}}_{\tau }[\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])](-t)\right)\,dt}{{\sqrt {2\pi }}-(2i)\left({\mathcal {F}}_{\tau }\left[{\frac {\exp({\text{CumulantGeneratingFunction}}[{\mathcal {D}},i\tau ])}{\tau }}\right](-x)\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccc6fce1513654e828f6ade9a41779fd4213f503)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/379d702fa951b8a1cbec6bee47e43a1ec76e804f)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9a0cdc0f60f3ab2f92dffd56eecc923b8cbae9)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]=\int _{0}^{\infty }(1-{\text{CDF}}[{\mathcal {D}},t]-{\text{CDF}}[{\mathcal {D}},-t])\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84a2e4dbeaf5a0598f1f2c876c32dd1a27388880)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]=-i{\text{CharacteristicFunction}}^{(0,1)}({\mathcal {D}},0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6acf7ba348d61d129b18ab790a59580ec5dd70d3)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]=-\int _{-\infty }^{\infty }t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b6f619c3c24962488098e16fd5e52340f7a5282)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]=\int _{0}^{\infty }({\text{SurvivalFunction}}[{\mathcal {D}},t]+{\text{SurvivalFunction}}[{\mathcal {D}},-t]-1)\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afe25c50f4c3e1f556d1fe3ca456fef4dc60f38a)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fe6c337b0bb900a1e5e2479f6d8317ee09f8da4)
![{\displaystyle {\text{Moment}}[{\mathcal {D}},1]={\text{Too complicated}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c7d49d466511b219beea4741b7be282cc0e659)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]=\int _{0}^{1}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8794370f94455296f3af310d86a5b44549ed5f4f)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]=\int _{0}^{1}{\text{Quantile}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52fcf067c96a64507c3418d8efc8431154d14a4e)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]=\int _{0}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7332d32bf86f1eab619a94e3ebf9f7005c5988e5)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{Median}}[{\mathcal {D}}]+\int _{0}^{1}\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},\tau )d\tau dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83816a58d11b0ad8a6bc42cc140c1f7b3b6a2b38)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t{\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t^{2}}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0083dd262c8a77d1232ea74ad8b4b140f90a66ca)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{Moment}}[{\mathcal {D}},1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1efd02b7d73c36fac6b0ef3a87091b64a2b840c7)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{FactorialMoment}}[{\mathcal {D}},1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98b16afa279dcab38d5c1ba6073e3399b7b80129)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{AscendingFactorialMoment}}({\mathcal {D}},1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41adee78ff1c19d2efa3671857145ae457a56c1c)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{Cumulant}}[{\mathcal {D}},1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9824a7e8d2982dfb5086b0545a044a4567707d51)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{FactorialCumulant}}({\mathcal {D}},1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89d83c8b5459717dbd35ac9db148a011bdd9172d)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{AscendingFactorialCumulant}}({\mathcal {D}},1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a53c95578f052bf04f5831c031e104cc945db8d)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{MomentGeneratingFunction}}^{(0,1)}({\mathcal {D}},0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9943be0fc283d2d8e28357780f23aef666f8c4)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{FactorialMomentGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3834a21d3856317d61a6e75418b9c2b0bd78482)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{DescendingFactorialMomentGeneratingFunction}}^{(0,1)}({\mathcal {D}},0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/582e3f3259ccd2345aada4f2a5c36d62673efcc8)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{AscendingFactorialMomentGeneratingFunction}}^{(0,1)}({\mathcal {D}},0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b53a175857007d575f251126a1f0838ca271cbc4)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{CumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/064819124d778df8edff8c46a2f2f16d919d7cd5)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ecb0bea012c234a636b862baf710f0ca84cce89)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{DescendingFactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d8864099c9b168625789a88eac0c236b98ce37)
![{\displaystyle {\text{Mean}}[{\mathcal {D}}]={\text{AscendingFactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfef638d5436b1d240661ea0e36fad25e420d111)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t^{2}{\text{PDF}}[{\mathcal {D}},t]\,dt-\left(\int _{-\infty }^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt\right){}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fa5f0039adb61d5a8769226fafab41323f09a7)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t^{2}{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt-\left(\int _{-\infty }^{\infty }t{\frac {\partial {\text{CDF}}[{\mathcal {D}},t]}{\partial t}}\,dt\right){}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bf97659f42aaaa7585dc3dd47ab8d41a79d4b91)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]={\text{CharacteristicFunction}}^{(0,1)}({\mathcal {D}},0)^{2}-{\text{CharacteristicFunction}}^{(0,2)}({\mathcal {D}},0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/157b04bca3a920b9392ed7bc638f86cda01e2301)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]=-\int _{-\infty }^{\infty }t^{2}{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt-\left(\int _{-\infty }^{\infty }t{\frac {\partial {\text{SurvivalFunction}}[{\mathcal {D}},t]}{\partial t}}\,dt\right){}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bedfd371b5a01207e59dea74941184c99f72f8fd)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }{\frac {t^{2}{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt-\left(\int _{-\infty }^{\infty }{\frac {t{\frac {\partial {\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\partial t}}}{e^{{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}}}\,dt\right){}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c98ea8d13fdef5bb3bd9f668644a9a4a183ec452)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]=\int _{0}^{1}{\text{InverseCDF}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt\right){}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d662c124478855d1aaa42c926ef43dda263267e)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]=\int _{0}^{1}{\text{Quantile}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{Quantile}}[{\mathcal {D}},t]\,dt\right){}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11606475876c82d2a78bbc2e40eab4a5ede22de7)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]=\int _{0}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt\right){}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2011ad40d11ef1d5a6c05250daa9f8820fd33d47)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]=\int _{0}^{1}\left(\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},\tau )\,d\tau \right){}^{2}\,dt-\left(\int _{0}^{1}\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},\tau )d\tau dt\right){}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d645d4f9649963e437c4bfb1b9380d6cd9e53e34)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]=\int _{-\infty }^{\infty }t^{2}{\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t^{2}}}\,dt-\left(\int _{-\infty }^{\infty }t{\frac {\partial ^{2}({\text{MeanResidualLifeFunction}}({\mathcal {D}},t){\text{SurvivalFunction}}[{\mathcal {D}},t])}{\partial t^{2}}}\,dt\right){}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27a510bf90b288e39f1a1e8c7e738341c5fbdb36)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]={\text{StandardDeviation}}[{\mathcal {D}}]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90a937796339a6d0c09c862eb0f4486bc01fed0d)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]={\text{Moment}}[{\mathcal {D}},2]-{\text{Moment}}[{\mathcal {D}},1]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a623fe7d79d687675f09ed28b40f95fe52708651)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]={\text{CentralMoment}}[{\mathcal {D}},2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2868d2760a87f51454655da4f53e4b617d8bf808)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]=-{\text{FactorialMoment}}[{\mathcal {D}},1]^{2}+{\text{FactorialMoment}}[{\mathcal {D}},1]+{\text{FactorialMoment}}[{\mathcal {D}},2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e1bf0e7e66e7dddff8a1799646d1660049a980)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]={\text{Cumulant}}[{\mathcal {D}},2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c2b01071e0f006f55a0c8916897536979ed7e2f)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]={\text{FactorialCumulant}}({\mathcal {D}},1)+{\text{FactorialCumulant}}({\mathcal {D}},2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9754d1d3e88cc63658a0d1f443f4e9c0e22beaa0)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]={\text{AscendingFactorialCumulant}}({\mathcal {D}},2)-{\text{AscendingFactorialCumulant}}({\mathcal {D}},1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5ad1331867d48ff2bdc69bb45ed70c558f1f89)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]={\text{CentralMomentGeneratingFunction}}^{(0,2)}({\mathcal {D}},0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c543c7bc0357e00915482ba780de4774930efa80)
![{\displaystyle {\text{Variance}}[{\mathcal {D}}]={\text{CumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd87d2b1345fa0147d40bca86d5f8ef5c67b51c)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{-\infty }^{\infty }t^{2}{\text{PDF}}[{\mathcal {D}},t]\,dt-\left(\int _{-\infty }^{\infty }t{\text{PDF}}[{\mathcal {D}},t]\,dt\right){}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/529d994c12d40ba7edc26945da19ecc1089e2c7d)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{-\infty }^{\infty }t^{2}{\text{CDF}}^{(0,1)}({\mathcal {D}},t)\,dt-\left(\int _{-\infty }^{\infty }t{\text{CDF}}^{(0,1)}({\mathcal {D}},t)\,dt\right){}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e51555ad15b9901b848bbd89438c883437258d7)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{CharacteristicFunction}}^{(0,1)}({\mathcal {D}},0)^{2}-{\text{CharacteristicFunction}}^{(0,2)}({\mathcal {D}},0)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1af15f4f5e0542486065612a8d84f2cb21ba8cef)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {-\int _{-\infty }^{\infty }t^{2}{\text{SurvivalFunction}}^{(0,1)}({\mathcal {D}},t)\,dt-\left(\int _{-\infty }^{\infty }t{\text{SurvivalFunction}}^{(0,1)}({\mathcal {D}},t)\,dt\right){}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6afde444cf4e05df72455ec407d612192e29fd0)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{-\infty }^{\infty }t^{2}e^{-{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\text{CumulativeHazardFunction}}^{(0,1)}({\mathcal {D}},t)\,dt-\left(\int _{-\infty }^{\infty }te^{-{\text{CumulativeHazardFunction}}({\mathcal {D}},t)}{\text{CumulativeHazardFunction}}^{(0,1)}({\mathcal {D}},t)\,dt\right){}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba0f31af09edad18c4bdf296d2103c424f93723)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{0}^{1}{\text{InverseCDF}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{InverseCDF}}[{\mathcal {D}},t]\,dt\right){}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f9f8788f00af1c4fe5b834aee596bcd1de44c05)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{0}^{1}{\text{Quantile}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{Quantile}}[{\mathcal {D}},t]\,dt\right){}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc9e4331c8e6f1f8d6925bf795ca18df4fcb255)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{0}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]^{2}\,dt-\left(\int _{0}^{1}{\text{InverseSurvivalFunction}}[{\mathcal {D}},t]\,dt\right){}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae0f26f17dc64ef84a9226d0110c1edbec0288a)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{0}^{1}\left(\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},\tau )\,d\tau \right){}^{2}\,dt-\left(\int _{0}^{1}\int _{\frac {1}{2}}^{t}{\text{Sparsity}}({\mathcal {D}},\tau )d\tau dt\right){}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac43e4fb06929e86a99afefbf43663304ae86bc)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {\int _{-\infty }^{\infty }t^{2}{\frac {\partial ^{2}({\text{SurvivalFunction}}[{\mathcal {D}},t]{\text{MeanResidualLifeFunction}}({\mathcal {D}},t))}{\partial t^{2}}}\,dt-\left(\int _{-\infty }^{\infty }t{\frac {\partial ^{2}({\text{SurvivalFunction}}[{\mathcal {D}},t]{\text{MeanResidualLifeFunction}}({\mathcal {D}},t))}{\partial t^{2}}}\,dt\right){}^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58c64f4fcc18622b7e7e48daf87d50f70749b5c6)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{Variance}}[{\mathcal {D}}]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b55cb7f6579799577c790358e9d81ae438df2dc2)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{Moment}}[{\mathcal {D}},2]-{\text{Moment}}[{\mathcal {D}},1]^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0da49ce35198ffa5a6fae0e121cf0926606f0d)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{CentralMoment}}[{\mathcal {D}},2]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc360fbd4c86b4ba82d95f625d55af20de3896d8)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{Cumulant}}[{\mathcal {D}},2]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94c431257cda9afdb264c1e5ca79288d5f2febe4)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{CentralMomentGeneratingFunction}}^{(0,2)}({\mathcal {D}},0)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7eb6e2e049c4d43c20a04c0ca9614bad97f36c6)
![{\displaystyle {\text{StandardDeviation}}[{\mathcal {D}}]={\sqrt {{\text{CumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f037e6cc8efb76eaa5a09c5c515f2330bcbf12c0)
![{\displaystyle {\text{Skewness}}[{\mathcal {D}}]={\frac {\int _{-\infty }^{\infty }(t-{\text{Mean}}[{\mathcal {D}}])^{3}{\frac {\partial ^{2}{\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}{\partial t^{2}}}\,dt}{{\text{StandardDeviation}}[{\mathcal {D}}]^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81f8ce9366fabdf26c70d67e84574c32f5838e35)
![{\displaystyle {\text{Skewness}}[{\mathcal {D}}]={\frac {{\text{CentralMoment}}[{\mathcal {D}},3]}{{\text{CentralMoment}}[{\mathcal {D}},2]^{3/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/159a1680333c295562125432f09552032bb57cff)
![{\displaystyle {\text{Skewness}}[{\mathcal {D}}]={\frac {{\text{Cumulant}}[{\mathcal {D}},3]}{{\text{Cumulant}}[{\mathcal {D}},2]^{3/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7d29384089960a852c74b800d701f6c4bc892b)
![{\displaystyle {\text{Skewness}}[{\mathcal {D}}]={\frac {{\text{FactorialCumulant}}({\mathcal {D}},1)+3{\text{FactorialCumulant}}({\mathcal {D}},2)+{\text{FactorialCumulant}}({\mathcal {D}},3)}{({\text{FactorialCumulant}}({\mathcal {D}},1)+{\text{FactorialCumulant}}({\mathcal {D}},2))^{3/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af98824baefe0ed2db6225ae5729170c484def4d)
![{\displaystyle {\text{Skewness}}[{\mathcal {D}}]={\frac {{\text{CumulantGeneratingFunction}}^{(0,3)}({\mathcal {D}},1)}{{\text{CumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)^{3/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2508a2fc31d8d9da8ff2db195f440dcf3b3e2408)
![{\displaystyle {\text{Skewness}}[{\mathcal {D}}]={\frac {{\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)+3{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)+{\text{FactorialCumulantGeneratingFunction}}^{(0,3)}({\mathcal {D}},1)}{\left({\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)+{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)\right)^{3/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d0829d6d02da3c9f40a01ae734b6fccf6101be4)
![{\displaystyle {\text{Kurtosis}}[{\mathcal {D}}]={\frac {\int _{-\infty }^{\infty }(t-{\text{Mean}}[{\mathcal {D}}])^{4}{\frac {\partial ^{2}{\text{MeanResidualLifeFunction}}({\mathcal {D}},t)}{\partial t^{2}}}\,dt}{{\text{StandardDeviation}}[{\mathcal {D}}]^{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97707d94530bcc2ab2e310f4d11aa4c74bf58d82)
![{\displaystyle {\text{Kurtosis}}[{\mathcal {D}}]={\frac {{\text{CentralMoment}}[{\mathcal {D}},4]}{{\text{CentralMoment}}[{\mathcal {D}},2]^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d33b6171f94db79b666218219ad183669b039e07)
![{\displaystyle {\text{Kurtosis}}[{\mathcal {D}}]=3+{\frac {{\text{Cumulant}}[{\mathcal {D}},4]}{{\text{Cumulant}}[{\mathcal {D}},2]^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a77d36c2d54e956c88744336783dd89316921c85)
![{\displaystyle {\text{Kurtosis}}[{\mathcal {D}}]=3+{\frac {{\text{FactorialCumulant}}({\mathcal {D}},1)+7{\text{FactorialCumulant}}({\mathcal {D}},2)+6{\text{FactorialCumulant}}({\mathcal {D}},3)+{\text{FactorialCumulant}}({\mathcal {D}},4)}{({\text{FactorialCumulant}}({\mathcal {D}},1)+{\text{FactorialCumulant}}({\mathcal {D}},2))^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6da6ac3524d1177b722c54191ff50ac0683bfc)
![{\displaystyle {\text{Kurtosis}}[{\mathcal {D}}]={\frac {3{\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)^{2}+7{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)+3{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)^{2}+{\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)\left(1+6{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)\right)+6{\text{FactorialCumulantGeneratingFunction}}^{(0,3)}({\mathcal {D}},1)+{\text{FactorialCumulantGeneratingFunction}}^{(0,4)}({\mathcal {D}},1)}{\left({\text{FactorialCumulantGeneratingFunction}}^{(0,1)}({\mathcal {D}},1)+{\text{FactorialCumulantGeneratingFunction}}^{(0,2)}({\mathcal {D}},1)\right)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0c79df5848702d8d25b4656acf0c42ecff2fb5)