# 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:

• $\star$

$a>0$

• $\star$

$b>0$

Syntax:

$X\sim Beta(a,b)$

Notes:

1. 1.

$X$ is used in many statistical models.

2. 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. 3.

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

4. 4.

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

5. 5.

$M_{X}(t)$ not useful

Title beta random variable BetaRandomVariable 2013-03-22 11:54:30 2013-03-22 11:54:30 mathcam (2727) mathcam (2727) 11 mathcam (2727) Definition msc 60-00 beta distribution