NegativeBinomialDistribution

The negative binomial distribution: the number of failures before the $n$-th success, with success probability $p$.

NegativeBinomialDistribution(n, p)an inert distribution object — carries $n, p$, unevaluated.

Typed signatures

Type	Package	Applies
(real<0..>, real<0..1>) -> distribution	statistics	always

Domain: Statistics

Details

• $PDF(k) = \binom{n+k-1}{k} p^n (1-p)^k$.
• $CDF(x) = I_p(n, \lfloor x\rfloor + 1)$, the regularized incomplete beta, via BetaRegularized — the standard negative-binomial/incomplete-beta identity — clamped to $0$ below $x=0$.
• $Mean = n(1-p)/p$, $Variance = n(1-p)/p^2$, both exact.
• RandomVariate sums $n$ GeometricDistribution-style draws.

Examples

Basic #

live ↗

In$\operatorname{PDF}(\mathrm{NegativeBinomialDistribution}(5, \frac{1}{3}), 2)$Out$\frac{20}{729}$

Cross-checked against wolframscript's N[CDF[NegativeBinomialDistribution[5, 1/3], 2]]

live ↗

In$\operatorname{CDF}(\mathrm{NegativeBinomialDistribution}(5, \frac{1}{3}), 2)$Out$\operatorname{BetaRegularized}(\frac{1}{3}, 5, 3)$

live ↗

In$\mathrm{mean}(\mathrm{NegativeBinomialDistribution}(5, \frac{1}{3}))$Out$10$

live ↗

In$\mathrm{Variance}(\mathrm{NegativeBinomialDistribution}(5, \frac{1}{3}))$Out$30$

Scope #

live ↗

In$\operatorname{CDF}(\mathrm{NegativeBinomialDistribution}(20, N(\frac{1}{3}, 30)), 5)$Out$0.000\,002\,268\,661\,443\,783\,180\,826\,88$

live ↗

In$\operatorname{PDF}(\mathrm{NegativeBinomialDistribution}(20, N(\frac{1}{3}, 30)), 5)$Out$0.000\,001\,605\,271\,196\,663\,597\,373\,910\,574\,262\,795\,274\,399\,330\,079\,52$

About

Examples	6
Checked against	Wolfram Language 2 agree

See also: GeometricDistribution, BinomialDistribution, PDF, CDF