Gamma distribution

Remarks

The mean is $\alpha / \beta$ and the variance is $\alpha / \beta^2$. Another common form uses the scale $\theta = 1/\beta$. $\operatorname{Gamma}(1, \beta)$ is the exponential distribution with rate $\beta$. For an integer $n \ge 1$, $\operatorname{Gamma}(n, \beta)$ is the law of a sum of $n$ independent $\operatorname{Exp}(\beta)$ random variables. It is the Erlang distribution. If $X \sim \operatorname{Gamma}(\alpha_1, \beta)$ and $Y \sim \operatorname{Gamma}(\alpha_2, \beta)$ are independent, then $X + Y \sim \operatorname{Gamma}(\alpha_1 + \alpha_2, \beta)$. The chi-squared distribution with $k$ degrees of freedom is $\operatorname{Gamma}(k/2, 1/2)$.