Relationship
Distribution
When
≡
{\displaystyle \equiv }
confluent hypergeometric
(
a
,
k
+
m
+
n
,
k
)
⋀
a
f
(
a
)
=
a
n
−
1
1
F
1
(
m
+
n
;
k
+
m
+
n
;
−
a
)
Γ
(
n
)
2
F
1
(
k
,
n
;
k
+
m
+
n
;
1
)
{\displaystyle (a,k+m+n,k)\quad {\underset {a}{\bigwedge }}\quad f(a)={\frac {a^{n-1}\,_{1}F_{1}(m+n;k+m+n;-a)}{\Gamma (n)\,_{2}F_{1}(k,n;k+m+n;1)}}}
≡
{\displaystyle \equiv }
negative binomial (k, p)
⋀
p
{\displaystyle \quad {\underset {p}{\bigwedge }}\quad }
beta (m,n)
m
,
n
>
0
{\displaystyle m,n>0}
≡
{\displaystyle \equiv }
Poisson
(
λ
)
⋀
λ
f
(
λ
)
=
e
λ
/
2
Γ
(
k
+
m
)
λ
1
2
(
k
+
n
−
3
)
Γ
(
m
+
n
)
W
1
2
(
−
k
−
n
+
1
)
−
m
,
n
−
k
2
(
λ
)
Γ
(
k
)
Γ
(
m
)
Γ
(
n
)
=
λ
n
−
1
Γ
(
k
+
m
)
Γ
(
m
+
n
)
U
(
m
+
n
,
−
k
+
n
+
1
,
λ
)
Γ
(
k
)
Γ
(
m
)
Γ
(
n
)
{\displaystyle (\lambda )\quad {\underset {\lambda }{\bigwedge }}\quad {\begin{aligned}f(\lambda )&={\frac {e^{\lambda /2}\Gamma (k+m)\lambda ^{{\frac {1}{2}}(k+n-3)}\Gamma (m+n)W_{{\frac {1}{2}}(-k-n+1)-m,{\frac {n-k}{2}}}(\lambda )}{\Gamma (k)\Gamma (m)\Gamma (n)}}\\&={\frac {\lambda ^{n-1}\Gamma (k+m)\Gamma (m+n)U(m+n,-k+n+1,\lambda )}{\Gamma (k)\Gamma (m)\Gamma (n)}}\end{aligned}}}
k
,
m
,
n
>
0
{\displaystyle k,m,n>0}
≡
{\displaystyle \equiv }
[Poisson (aj)
⋀
a
{\displaystyle {\underset {a}{\bigwedge }}}
gamma (k,1) ]
⋀
j
f
(
j
)
=
j
m
−
1
B
(
m
,
n
)
(
1
+
j
)
m
+
n
{\displaystyle {\underset {j}{\bigwedge }}f(j)={\frac {j^{m-1}}{B(m,n){(1+j)}^{m+n}}}}
j
≥
0
{\displaystyle j\geq 0}
⇐
{\displaystyle \Leftarrow }
Holla-negative binomial
(
a
,
b
,
k
,
β
)
{\displaystyle (a,b,k,\beta )}
a
=
m
b
=
n
β
=
0
{\displaystyle a=m\qquad b=n\qquad \beta =0}
⇐
{\displaystyle \Leftarrow }
inverse-Pólya
(
k
,
m
,
n
,
r
)
{\displaystyle (k,m,n,r)}
r
=
1
{\displaystyle r=1}
⇐
{\displaystyle \Leftarrow }
Kemp-Dacey-hypergeometric family
(
I
/
8
)
{\displaystyle (I/8)}
⇐
{\displaystyle \Leftarrow }
Ord family
(
a
,
b
0
,
b
1
,
b
2
)
{\displaystyle (a,b_{0},b_{1},b_{2})}
a
=
(
k
−
1
)
(
n
−
1
)
(
m
+
1
)
b
0
=
0
b
1
=
k
+
m
+
n
m
+
1
b
2
=
1
m
+
1
{\displaystyle a={\frac {(k-1)(n-1)}{(m+1)}}\qquad b_{0}=0\qquad b_{1}={\frac {k+m+n}{m+1}}\qquad b_{2}={\frac {1}{m+1}}}
⇒
{\displaystyle \Rightarrow }
deterministic (0)
n
=
0
{\displaystyle n=0}
⇒
{\displaystyle \Rightarrow }
1-shifted Feller-Shreve
{
n
=
1
k
=
m
=
1
2
k
=
1
m
=
n
=
1
2
{\displaystyle {\begin{cases}n=1\qquad k=m={\frac {1}{2}}\\k=1\qquad m=n={\frac {1}{2}}\\\end{cases}}}
⇒
{\displaystyle \Rightarrow }
1-shifted Johnson-Kotz (a)
{
n
=
m
=
1
k
=
a
k
=
m
=
1
n
=
a
{\displaystyle {\begin{cases}n=m=1\qquad k=a\\k=m=1\qquad n=a\\\end{cases}}}
⇒
{\displaystyle \Rightarrow }
Kemp family Type IV
(
a
,
b
,
n
′
)
{\displaystyle (a,b,n')}
k
=
−
a
m
=
a
+
b
+
1
n
=
n
′
(
k
,
m
,
n
)
>
0
{\displaystyle k=-a\qquad m=a+b+1\qquad n=n'\qquad (k,m,n)>0}
⇒
{\displaystyle \Rightarrow }
Marlow
(
m
′
,
n
′
)
{\displaystyle (m',n')}
k
=
m
′
−
n
′
+
1
m
=
n
′
−
1
n
=
1
k
,
m
∈
N
{\displaystyle k=m'-n'+1\qquad m=n'-1\qquad n=1\qquad {k,m}\in \mathbb {N} }
⇒
{\displaystyle \Rightarrow }
1-shifted Miller
(
m
′
)
{\displaystyle (m')}
k
=
1
m
=
m
′
+
1
n
=
1
{\displaystyle k=1\qquad m=m'+1\qquad n=1}
⇒
{\displaystyle \Rightarrow }
1-shifted Prasad
(
k
)
{\displaystyle (k)}
n
=
1
m
=
2
{\displaystyle n=1\qquad m=2}
⇒
{\displaystyle \Rightarrow }
Salvia-Bolinger
(
α
)
{\displaystyle (\alpha )}
k
=
1
m
=
α
n
=
1
−
α
{\displaystyle k=1\qquad m=\alpha \qquad n=1-\alpha }
⇒
{\displaystyle \Rightarrow }
1-shifted Schwarz-Tversky 1
k
=
n
=
1
m
=
2
{\displaystyle k=n=1\qquad m=2}
⇒
{\displaystyle \Rightarrow }
1-shifted Simon
k
=
1
m
=
1
n
=
1
{\displaystyle k=1\qquad m=1\qquad n=1}
⇒
{\displaystyle \Rightarrow }
Waring
(
b
,
n
)
{\displaystyle (b,n)}
k
=
1
m
=
b
{\displaystyle k=1\qquad m=b}
⇒
{\displaystyle \Rightarrow }
Yule
(
m
)
{\displaystyle (m)}
k
=
1
n
=
1
{\displaystyle k=1\qquad n=1}
→
{\displaystyle \rightarrow }
2-shifted Flory
k
=
2
m
→
∞
n
→
∞
m
m
+
n
→
1
2
{\displaystyle k=2\qquad m\rightarrow \infty \qquad n\rightarrow \infty \qquad {\frac {m}{m+n}}\rightarrow {\frac {1}{2}}}
→
{\displaystyle \rightarrow }
geometric (q)
n
=
1
k
→
∞
m
→
∞
m
m
+
n
→
p
{\displaystyle n=1\qquad k\rightarrow \infty \qquad m\rightarrow \infty \qquad {\frac {m}{m+n}}\rightarrow p}
→
{\displaystyle \rightarrow }
negative binomial (k,p)
m
→
∞
n
→
∞
m
m
+
n
→
p
{\displaystyle m\rightarrow \infty \qquad n\rightarrow \infty \qquad {\frac {m}{m+n}}\rightarrow p}
→
{\displaystyle \rightarrow }
Poisson (a)
m
→
∞
n
→
∞
k
→
∞
n
m
+
n
→
0
k
n
m
+
n
→
a
{\displaystyle m\rightarrow \infty \qquad n\rightarrow \infty \qquad k\rightarrow \infty \qquad {\frac {n}{m+n}}\rightarrow 0\qquad {\frac {kn}{m+n}}\rightarrow a}