beta random variable

$X$ is a beta random variable with parameters a and b if

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

Parameters:

• $a>0$

• $b>0$

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 $\beta (a,b)=\frac{\mathrm{\Gamma }(a)\mathrm{\Gamma }(b)}{\mathrm{\Gamma }(a+b)}$ (For information on the $\mathrm{\Gamma }$ 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