Definition:Beta Distribution

Definition
Let $X$ be a continuous random variable on a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

Let $\operatorname{Im} \left({X}\right) = \left[{0 \,.\,.\, 1}\right]$.

$X$ is said to have a Beta distribution if it has probability density function:


 * $\displaystyle f_X \left({x}\right) = \frac{ x^{\alpha - 1} \left({1 - x}\right)^{\beta - 1} } {\Beta \left({\alpha, \beta}\right)}$

for $\alpha, \beta > 0$, where $\Beta$ is the Beta function.

This is written:


 * $X \sim \operatorname{Beta} \left({\alpha, \beta}\right)$