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Student[Statistics][BetaRandomVariable] - beta random variable

 Calling Sequence BetaRandomVariable(nu, omega)

Parameters

 nu - first shape parameter omega - second shape parameter

Description

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

$f\left(t\right)=\left\{\begin{array}{cc}0& t<0\\ \frac{{t}^{-1+\mathrm{\nu }}{\left(1-t\right)}^{-1+\mathrm{\omega }}}{\mathrm{Β}\left(\mathrm{\nu },\mathrm{\omega }\right)}& t<1\\ 0& \mathrm{otherwise}\end{array}\right\$

 subject to the following conditions:

$0<\mathrm{\nu },0<\mathrm{\omega }$

 • 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)).

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):$
 > $X:=\mathrm{BetaRandomVariable}\left(\mathrm{ν},\mathrm{ω}\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 ${{}\begin{array}{cc}{0}& {u}{<}{0}\\ \frac{{{u}}^{{\mathrm{ν}}{-}{1}}{}{\left({1}{-}{u}\right)}^{{\mathrm{ω}}{-}{1}}}{{\mathrm{Β}}{}\left({\mathrm{ν}}{,}{\mathrm{ω}}\right)}& {u}{<}{1}\\ {0}& {\mathrm{otherwise}}\end{array}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{{0.5}}^{{\mathrm{ν}}{-}{1.}}{}{{0.5}}^{{-}{1.}{+}{\mathrm{ω}}}}{{\mathrm{Β}}{}\left({\mathrm{ν}}{,}{\mathrm{ω}}\right)}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 $\frac{{\mathrm{ν}}}{{\mathrm{ν}}{+}{\mathrm{ω}}}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\frac{{\mathrm{ν}}{}{\mathrm{ω}}}{{\left({\mathrm{ν}}{+}{\mathrm{ω}}\right)}^{{2}}{}\left({\mathrm{ν}}{+}{\mathrm{ω}}{+}{1}\right)}$ (4)
 > $Y:=\mathrm{BetaRandomVariable}\left(4,7\right):$
 > $\mathrm{PDF}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{CDF}\left(Y,x\right)$
 ${{}\begin{array}{cc}{0}& {x}{<}{0}\\ {210}{}{{x}}^{{4}}{}{\mathrm{hypergeom}}{}\left(\left[{-}{6}{,}{4}\right]{,}\left[{5}\right]{,}{x}\right)& {x}{<}{1}\\ {1}& {\mathrm{otherwise}}\end{array}$ (5)
 > $\mathrm{CDF}\left(Y,0.5,\mathrm{output}=\mathrm{plot}\right)$