GammaRandomVariable - Maple Help

Student[Statistics]

 GammaRandomVariable
 gamma random variable

 Calling Sequence GammaRandomVariable(b, c)

Parameters

 b - scale parameter c - shape parameter

Description

 • The gamma random variable is a continuous probability random variable with probability density function given by:

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ \frac{{\left(\frac{t}{b}\right)}^{c-1}{ⅇ}^{-\frac{t}{b}}}{b\mathrm{\Gamma }\left(c\right)}& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0

 • The gamma variate with scale parameter b and shape parameter 1 is equivalent to the Exponential variate with scale parameter b.
 • The gamma variate with scale parameter 1 and shape parameter c is equivalent to the Erlang variate with shape parameter c.

Examples

 > $\mathrm{with}\left({\mathrm{Student}}_{\mathrm{Statistics}}\right):$
 > $X≔\mathrm{GammaRandomVariable}\left(b,c\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\left\{\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{\left(\frac{{u}}{{b}}\right)}^{{c}{-}{1}}{}{{ⅇ}}^{{-}\frac{{u}}{{b}}}}{{b}{}{\mathrm{\Gamma }}{}\left({c}\right)}& {\mathrm{otherwise}}\end{array}\right\$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{\left(\frac{{0.5}}{{b}}\right)}^{{c}{-}{1.}}{}{{ⅇ}}^{{-}\frac{{0.5}}{{b}}}}{{b}{}{\mathrm{\Gamma }}{}\left({c}\right)}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${b}{}{c}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 ${{b}}^{{2}}{}{c}$ (4)
 > $Y≔\mathrm{GammaRandomVariable}\left(2,3\right):$
 > $\mathrm{PDF}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{CDF}\left(Y,x\right)$
 $\left\{\begin{array}{cc}{0}& {x}{\le }{0}\\ {-}\frac{{{x}}^{{2}}{}{{ⅇ}}^{{-}\frac{{x}}{{2}}}}{{8}}{-}\frac{{{ⅇ}}^{{-}\frac{{x}}{{2}}}{}{x}}{{2}}{-}{{ⅇ}}^{{-}\frac{{x}}{{2}}}{+}{1}& {0}{<}{x}\end{array}\right\$ (5)
 > $\mathrm{CDF}\left(Y,6,\mathrm{output}=\mathrm{plot}\right)$

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][GammaRandomVariable] command was introduced in Maple 18.