negative binomial distribution

Key facts

Notation

N B ( r , p ) {\displaystyle \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 ↦ ( 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

Cdf

k ↦ I p ( r , k + 1 ) , {\displaystyle k\mapsto I_{p}(r,\,k+1),} the regularized incomplete beta function

Mean

r ( 1 − p ) p {\displaystyle {\frac {r(1-p)}{p}}}

Mode

1 \\\\\n 0 & \\text{if } r\\leq 1\\end{cases}"}}'> { ⌊ ( 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}}}

Variance

r ( 1 − p ) p 2 {\displaystyle {\frac {r(1-p)}{p^{2}}}}

Skewness

2 − p ( 1 − p ) r {\displaystyle {\frac {2-p}{\sqrt {(1-p)r}}}}

Excess kurtosis

6 r + p 2 ( 1 − p ) r {\displaystyle {\frac {6}{r}}+{\frac {p^{2}}{(1-p)r}}}

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)}

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} }

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}}}

Fisher information

r p 2 ( 1 − p ) {\displaystyle {\frac {r}{p^{2}(1-p)}}}

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]}}}

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Wikidata facts

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