beta random variable

$f_{X}(x)=\frac{x^{a-1}(1-x)^{b-1}}{\beta(a,b)}$ , $x\in[0,1]$

Parameters:

Syntax:

$X\sim Beta(a,b)$

Notes:

1.

$X$ is used in many statistical models.
2.

The function $\beta:R\times R\to R$ is defined as $\beta(a,b)=\int_{0}^{1}{x^{a-1}(1-x)^{b-1}dx}$ . $\beta(a,b)$ can be calculated as $\beta(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$ (For information on the $\Gamma$ function, see the gamma random variable)
3.

$E[X]=\frac{a}{a+b}$
4.

$Var[X]=\frac{ab}{(a+b+1)(a+b)^{2}}$
5.

$M_{X}(t)$ not useful