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The negative binomial distribution is a discrete probability distribution with probability function given by:

subject to the following conditions:

The negative binomial distribution is also known as the Pascal distribution.

Note that the NegativeBinomial command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{NegativeBinomial}\left(x\,p\right)\right)\:$

$\mathrm{ProbabilityFunction}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \left(\genfrac{}{}{0ex}{}{x+u-1}{u}\right)p^x{\left(1-p\right)}^u& \mathrm{otherwise}\end{array}\right.$

$\mathrm{ProbabilityFunction}\left(X\&comma;2\right)$

$\left(\genfrac{}{}{0ex}{}{x+1}{2}\right)p^x{\left(1-p\right)}^2$

$\mathrm{Mean}\left(X\right)$

$\frac{x\left(1-p\right)}{p}$

$\mathrm{Variance}\left(X\right)$

$\frac{x\left(1-p\right)}{p^2}$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.