Definition:Gamma Distribution

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

Let $\Img X = \R_{\ge 0}$.

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


 * $\map {f_X} x = \dfrac {\beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\map \Gamma \alpha}$

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

This is written:


 * $X \sim \map \Gamma {\alpha, \beta}$