Beta Distribution

A continuous r.v. $X$ is said to have the Beta Distribution with parameters $a$ and $b$, where $a,b>0$, if its PDF is

$$f(x)=\frac{1}{\beta (a,b)}{x}^{a-1}(1-x{)}^{b-1},0<x<1$$

where the constant $\beta (a,b)$ is chosen to make the PDF integrate to $1$. We write this as $X\sim \mathrm{Beta}(a,b)$

By definition, the constant $\beta (a,b)$ satisfies

$$\beta (a,b)={\int }_0^1{x}^{a-1}(1-x{)}^{b-1}\mathrm{d}x$$

which is the beta function

The expectation of beta distribution is $\mathrm{E}[X]=\frac{a}{a+b}$