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Student[Statistics]

 NegativeBinomialRandomVariable
 negative binomial (Pascal) random variable

 Calling Sequence NegativeBinomialRandomVariable(x, p)

Parameters

 x - number of trials p - probability of success

Description

 • The negative binomial random variable is a discrete probability random variable 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 random variable is also known as the Pascal random variable.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):$
 > $X≔\mathrm{NegativeBinomialRandomVariable}\left(x,p\right):$
 > $\mathrm{ProbabilityFunction}\left(X,u\right)$
 ${{}\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{\mathrm{Γ}}{}\left({x}{+}{u}\right){}{{p}}^{{x}}{}{\left({1}{-}{p}\right)}^{{u}}}{{\mathrm{Γ}}{}\left({x}\right){}{u}{!}}& {\mathrm{otherwise}}\end{array}$ (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)
 > $Y≔\mathrm{NegativeBinomialRandomVariable}\left(3,\frac{1}{2}\right):$
 > $\mathrm{ProbabilityFunction}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{CDF}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)$

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.

Compatibility

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