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Statistics[Distributions]

 NegativeBinomial
 negative binomial (Pascal) distribution

 Calling Sequence NegativeBinomial(x, p) NegativeBinomialDistribution(x, p)

Parameters

 x - number of trials p - probability of success

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ \frac{\mathrm{\Gamma }\left(x+t\right){p}^{x}{\left(1-p\right)}^{t}}{\mathrm{\Gamma }\left(x\right)t!}& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{NegativeBinomial}\left(x,p\right)\right):$
 > $\mathrm{ProbabilityFunction}\left(X,u\right)$
 ${{}\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{\mathrm{Γ}}{}\left({x}{+}{u}\right){}{{p}}^{{x}}{}{\left({1}{-}{p}\right)}^{{u}}}{{\mathrm{Γ}}{}\left({x}\right){}{u}{!}}& {\mathrm{otherwise}}\end{array}$ (1)
 > $\mathrm{ProbabilityFunction}\left(X,2\right)$
 $\frac{{1}}{{2}}{}\frac{{\mathrm{Γ}}{}\left({x}{+}{2}\right){}{{p}}^{{x}}{}{\left({1}{-}{p}\right)}^{{2}}}{{\mathrm{Γ}}{}\left({x}\right)}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 $\frac{{x}{}\left({1}{-}{p}\right)}{{p}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\frac{{x}{}\left({1}{-}{p}\right)}{{{p}}^{{2}}}$ (4)

References

 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.