| Negative binomial distribution | CDF | k ↦ I p ( r , k + 1 ) , {\displaystyle k\mapsto I_{p}(r,\,k+1),} the regularized incomplete beta function | 1.00 | infobox |
| Negative binomial distribution | CF | ( p 1 − ( 1 − p ) e i t ) r with t ∈ R {\displaystyle {\biggl (}{\frac {p}{1-(1-p)e^{i\,t}}}{\biggr )}^{\!r}{\text{ with }}t\in \mathbb {R} } | 1.00 | infobox |
| Negative binomial distribution | Excess kurtosis | 6 r + p 2 ( 1 − p ) r {\displaystyle {\frac {6}{r}}+{\frac {p^{2}}{(1-p)r}}} | 1.00 | infobox |
| Negative binomial distribution | Fisher information | r p 2 ( 1 − p ) {\displaystyle {\frac {r}{p^{2}(1-p)}}} | 1.00 | infobox |
| Negative binomial distribution | Mean | r ( 1 − p ) p {\displaystyle {\frac {r(1-p)}{p}}} | 1.00 | infobox |
| Negative binomial distribution | Method of moments | r = E [ X ] 2 V [ X ] − E [ X ] {\displaystyle r={\frac {E[X]^{2}}{V[X]-E[X]}}} p = E [ X ] V [ X ] {\displaystyle p={\frac {E[X]}{V[X]}}} | 1.00 | infobox |
| Negative binomial distribution | MGF | ( p 1 − ( 1 − p ) e t ) r for t < − log ( 1 − p ) {\displaystyle {\biggl (}{\frac {p}{1-(1-p)e^{t}}}{\biggr )}^{\!r}{\text{ for }}t<-\log(1-p)} | 1.00 | infobox |
| Negative binomial distribution | Mode | { ⌊ ( r − 1 ) ( 1 − p ) p ⌋ if r > 1 0 if r ≤ 1 {\displaystyle {\begin{cases}\left\lfloor {\frac {(r-1)(1-p)}{p}}\right\rfloor &{\text{if }}r>1\\0&{\text{if }}r\leq 1\end{cases}}} | 1.00 | infobox |
| Negative binomial distribution | Notation | N B ( r , p ) {\displaystyle \mathrm {NB} (r,\,p)} | 1.00 | infobox |
| Negative binomial distribution | 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) | 1.00 | infobox |
| Negative binomial distribution | PGF | ( p 1 − ( 1 − p ) z ) r for | z | < 1 1 − p {\displaystyle {\biggl (}{\frac {p}{1-(1-p)z}}{\biggr )}^{\!r}{\text{ for }}|z|<{\frac {1}{1-p}}} | 1.00 | infobox |
| Negative binomial distribution | PMF | k ↦ ( k + r − 1 k ) ⋅ ( 1 − p ) k p r , {\displaystyle k\mapsto {k+r-1 \choose k}\cdot (1-p)^{k}p^{r},} involving a binomial coefficient | 1.00 | infobox |
| Negative binomial distribution | Skewness | 2 − p ( 1 − p ) r {\displaystyle {\frac {2-p}{\sqrt {(1-p)r}}}} | 1.00 | infobox |
| Negative binomial distribution | Support | k ∈ { 0, 1, 2, 3, … } — number of failures | 1.00 | infobox |
| Negative binomial distribution | Variance | r ( 1 − p ) p 2 {\displaystyle {\frac {r(1-p)}{p^{2}}}} | 1.00 | infobox |
| Negative binomial distribution | is a | special case of the discrete phase-type distribution.The negative binomial distribution is a special case of discrete compound Poisson distribution.Poisson distributionConsider… | 0.90 | text |
