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

subject to the following conditions:

The negative binomial random variable is also known as the Pascal random variable.

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

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

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

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

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

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

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

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

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

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

$Y≔\mathrm{NegativeBinomialRandomVariable}\left(3\&comma;\frac{1}{2}\right)\&colon;$

$\mathrm{ProbabilityFunction}\left(Y\&comma;x\&comma;\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{CDF}\left(Y\&comma;x\&comma;\mathrm{output}=\mathrm{plot}\right)$

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.

The Student[Statistics][NegativeBinomialRandomVariable] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.