Template:Negative binomial distribution

Different texts (and even different parts of this article) adopt slightly different definitions for the negative binomial distribution. They can be distinguished by whether the support starts at k = 0 or at k = r, whether p denotes the probability of a success or of a failure, and whether r represents success or failure, so identifying the specific parametrization used is crucial in any given text.
	Probability mass function; The orange line represents the mean, which is equal to 10 in each of these plots; the green line shows the standard deviation.
Notation
Parameters	r > 0 — number of successes until the experiment is stopped (integer, but the definition can also be extended to reals); p ∈ [0,1] — success probability in each experiment (real)
Support	k ∈ { 0, 1, 2, 3, … } — number of failures
PMF	involving a binomial coefficient
CDF	the regularized incomplete beta function
Mean
Mode
Variance
Skewness
Excess kurtosis
MGF
CF
PGF
Fisher information
Method of moments	;

NOTE: This list only appears on this page, and will not be included in the template.

^ DeGroot, Morris H. (1986). Probability and Statistics (Second ed.). Addison-Wesley. pp. 258–259. ISBN 0-201-11366-X. LCCN 84006269. OCLC 10605205.

[DeGrootNB-1] DeGroot, Morris H. (1986). Probability and Statistics (Second ed.). Addison-Wesley. pp. 258–259. ISBN 0-201-11366-X. LCCN 84006269. OCLC 10605205.

[1]

Different texts (and even different parts of this article) adopt slightly different definitions for the negative binomial distribution. They can be distinguished by whether the support starts at k = 0 or at k = r, whether p denotes the probability of a success or of a failure, and whether r represents success or failure,^[1] so identifying the specific parametrization used is crucial in any given text.
Probability mass function The orange line represents the mean, which is equal to 10 in each of these plots; the green line shows the standard deviation.
Notation	$\mathrm {NB} (r,\,p)$
Parameters	r > 0 — number of successes until the experiment is stopped (integer, but the definition can also be extended to reals) p ∈ [0,1] — success probability in each experiment (real)
Support	k ∈ { 0, 1, 2, 3, … } — number of failures
PMF	$k\mapsto {k+r-1 \choose k}\cdot (1-p)^{k}p^{r},$ involving a binomial coefficient
CDF	$k\mapsto I_{p}(r,\,k+1),$ the regularized incomplete beta function
Mean	${\frac {r(1-p)}{p}}$
Mode	${\begin{cases}\left\lfloor {\frac {(r-1)(1-p)}{p}}\right\rfloor &{\text{if }}r>1\\0&{\text{if }}r\leq 1\end{cases}}$
Variance	${\frac {r(1-p)}{p^{2}}}$
Skewness	${\frac {2-p}{\sqrt {(1-p)r}}}$
Excess kurtosis	${\frac {6}{r}}+{\frac {p^{2}}{(1-p)r}}$
MGF	${\biggl (}{\frac {p}{1-(1-p)e^{t}}}{\biggr )}^{\!r}{\text{ for }}t<-\log(1-p)$
CF	${\biggl (}{\frac {p}{1-(1-p)e^{i\,t}}}{\biggr )}^{\!r}{\text{ with }}t\in \mathbb {R}$
PGF	${\biggl (}{\frac {p}{1-(1-p)z}}{\biggr )}^{\!r}{\text{ for }}\|z\|<{\frac {1}{p}}$
Fisher information	${\frac {r}{p^{2}(1-p)}}$
Method of moments	$r={\frac {E[X]^{2}}{V[X]-E[X]}}$ $p={\frac {E[X]}{V[X]}}$