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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.

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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	$r={\frac {E[X]^{2}}{V[X]-E[X]}}$ $p={\frac {E[X]}{V[X]}}$