Negative Binomial Distribution (NB₍r,p₎) ○◂｜_append｜___｜20260628182425-01-•

A	B	C	D
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	$k\mapsto \left(\genfrac{}{}{0ex}{}{k+r-1}{k}\right)\cdot (1-p{)}^k{p}^r,$ involving a binomial coefficient	…	…
CDF	$k\mapsto I_p(r,k+1),$ the regularized incomplete beta function	…	…
Mean	$\frac{r(1-p)}{p}$	…	…
Mode	$\{\begin{array}{ll}\lfloor \frac{(r-1)(1-p)}{p}\rfloor & \text{if }r>1\\ 0& \text{if }r\le 1\end{array}$	…	…
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	$(\frac{p}{1-(1-p)e^t}{)}^r\text{ for }t<-\log (1-p)$	…	…
CF	$(\frac{p}{1-(1-p)e^{it}}{)}^r\text{ with }t\in \mathbb{R}$	…	…
PGF	$(\frac{p}{1-(1-p)z}{)}^r\text{ for }｜z｜<\frac{1}{1-p}$	…	…
Fisher information	$\frac{r}{p^2(1-p)}$	…	…
Method of moments	$r=\frac{E[X{]}^2}{V[X]-E[X]}$ $p=\frac{E[X]}{V[X]}$	…	…

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Footnotes

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