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

subject to the following conditions:

The beta distribution 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)).

Note that the Beta(a, b) returns the value of the Beta function with parameters a and b, so in order to define a Beta random variable one should use the unevaluated name 'Beta'. In 2D math notation, the capital letter $\mathrm{{\rm B}}$ looks like a capital letter $B$, but the two are different in Maple.

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

$\mathrm{RandomVariable}\left(\mathrm{{\rm B}}\left(1\,2\right)\right)$

Error, (in Statistics:-Distribution) invalid input: too many and/or wrong type of arguments passed to Statistics:-Distributions:-DataStructure:-NewDistribution; first unused argument is 1/2

$\mathrm{RandomVariable}\left('\mathrm{{\rm B}}'\left(1\,2\right)\right)$

$\mathrm{\_R}$

$\mathrm{RandomVariable}\left(\mathrm{BetaDistribution}\left(1\,2\right)\right)$

$\mathrm{\_R0}$

$X≔\mathrm{RandomVariable}\left('\mathrm{{\rm B}}'\left(\mathrm{\nu }\,\mathrm{\omega }\right)\right)\:$

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

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

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

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

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

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

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

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

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.