> [!tldr] > The **gamma distribution** $\mathrm{Gamma}(\alpha, \beta)$ with parameters $\text{shape}=\alpha>0$, $\text{rate}=\beta>0$ has density function $f_{\alpha,\beta}(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x},\,\,x >0,$where the **gamma function** $\Gamma(\alpha)=(\alpha-1)!$ when $\alpha$ is an integer. > > Gamma can be [[Bayesian Inference#Conjugation in Exponential Families|conjugate]] priors of rate-like parameters like $\lambda$ in exponential $\mathrm{Exp}(\lambda)$, chi-squared, and gamma likelihoods. If $X \sim \mathrm{Gamma}(\alpha,\beta)$, then its $k$th moment is $\mathbb{E}[X^{k}]=\frac{\Gamma(\alpha+k)}{\Gamma(\alpha)\beta^{k}},$and in particular its first two moments are $\begin{align*} \mathbb{E}[X^{1}]&= \alpha / \beta,\\ \mathbb{E}[X^{2}]&= \alpha(\alpha+1) / \beta^{2}. \end{align*}$Therefore it has variance $\mathrm{Var}(X)=\mathbb{E}[X^{2}]-\mathbb{E}[X]^{2}=\alpha / \beta^{2}.$ ### Properties of Gamma Distribution - *Sum of exponentials is gamma*: if $\alpha$ is an integer, $X_{1},\dots,X_{\alpha} \overset{\mathrm{iid.}}{\sim}\exp(\beta)$, then $\sum_{i}X_{i} \sim \Gamma(\alpha,\beta)$. In particular, $\Gamma(1,\beta)=\exp(\beta)$. - *Chi-squared is gamma*, in particular $\chi^{2}_{k} = \mathrm{Gamma}(k / 2, 1 / 2)$. - Furthermore, $X_{i} / \sum_{j}X_{j} \sim \mathrm{Pareto}(1,n)-1$.