Gamma Distribution

Definition

$X\sim \Gamma (\alpha ,\beta )=\Gamma (k,\theta )$ where $\alpha =k\in {\mathbb{R}}^{+}$ is the shape parameter, and $\beta =\frac{1}{\theta }\in {\mathbb{R}}^{+}$ is the scale parameter

Distribution that models the waiting time until occurring $\alpha$th events in a Poisson process with mean $\theta =\frac{1}{\beta }$

Properties

PDF

$f(x)=\frac{x^{k-1}{e}^{-\frac{x}{\theta }}}{\Gamma (k){\theta }^k}=\frac{{\beta }^{\alpha }{x}^{\alpha -1}{e}^{-\beta x}}{\Gamma (\alpha )},x\in {\mathbb{R}}^{+}$ where $\Gamma (\alpha )$ is the Gamma Function

Mean

$E(X)=k\theta =\frac{\alpha }{\beta }$

Variance

$\mathrm{Var}(X)=k{\theta }^2=\frac{\alpha }{{\beta }^2}$

MGF

$M(t)=(1-\theta t{)}^{-k}={\left(1-\frac{t}{\beta }\right)}^{-\alpha }$ where $t<\frac{1}{\theta }$ and $t<\beta$

Sum of independent Gamma distributions

Let $X_i\sim \Gamma ({\alpha }_i,\beta )$‘s are independent gamma distributions $\sum _{i=1}^n{X}_i\sim \Gamma \left(\sum _{i=1}^n{\alpha }_i,\beta \right)$

Scaling

Let $X\sim \Gamma (k,\theta )$, then, $\forall c>0,cX\sim \Gamma (k,c\theta )$

Let $X\sim \Gamma (\alpha ,\beta )$, then, $\forall c>0,cX\sim \Gamma (\alpha ,\frac{\beta }{c})$

Relationship with Poisson Distribution

Let $W$ be the time until the $k$-th event in a poisson process with rate $\lambda$, then, the cdf and pdf of $W$ is $G(w)={\int }_0^w\frac{{\lambda }^k{y}^{k-1}{e}^{-\lambda y}}{\Gamma (k)}dy,w\in {\mathbb{R}}^{+}$ $g(w)=\frac{{\lambda }^k{w}^{k-1}{e}^{-\lambda w}}{\Gamma (k)}\sim \Gamma \left(k,\frac{1}{\lambda }\right),w\in {\mathbb{R}}^{+}$