The beta distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The beta random variable is related to the independent Gamma variates Gamma(1,nu) and Gamma(1,omega) by the formula Beta(nu,omega) ~ Gamma(1,nu)/(Gamma(1,nu)+Gamma(1,omega)).

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

$X≔\mathrm{BetaRandomVariable}\left(\mathrm{\nu }\,\mathrm{\omega }\right)\:$

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

$\left\{\begin{array}{cc}0& u<0\\ \frac{u^{-1+\mathrm{\nu }}{\left(1-u\right)}^{-1+\mathrm{\omega }}}{\mathrm{{\rm B}}\left(\mathrm{\nu }\&comma;\mathrm{\omega }\right)}& u<1\\ 0& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;0.5\right)$

$\frac{{0.5}^{-1.+\mathrm{\nu }}{0.5}^{-1.+\mathrm{\omega }}}{\mathrm{{\rm B}}\left(\mathrm{\nu }\&comma;\mathrm{\omega }\right)}$

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

$\frac{\mathrm{\nu }}{\mathrm{\nu }+\mathrm{\omega }}$

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

$\frac{\mathrm{\nu }\mathrm{\omega }}{{\left(\mathrm{\nu }+\mathrm{\omega }\right)}^2\left(\mathrm{\nu }+\mathrm{\omega }+1\right)}$

$Y≔\mathrm{BetaRandomVariable}\left(4\&comma;7\right)\&colon;$

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

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

$\left\{\begin{array}{cc}0& x<0\\ 210x^4\mathrm{hypergeom}\left(\left[\mathrm{-6}\&comma;4\right]\&comma;\left[5\right]\&comma;x\right)& x<1\\ 1& \mathrm{otherwise}\end{array}\right.$

$\mathrm{CDF}\left(Y\&comma;0.5\&comma;\mathrm{output}=\mathrm{plot}\right)$

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][BetaRandomVariable] command was introduced in Maple 18.

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