Definition:Gamma Distribution

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

Let $\operatorname{Im} \left({X}\right) = \R_{\ge 0}$.

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


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

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

This is written:


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