Gamma Distribution

def. Gamma Distribution. In a Poisson Point Process with intensity $\lambda$, let $W_i$ be the “wait times” between the $i-1$ -th and $i\text{-th}$ event. let $X$ be the total “wait time” for $r$ events; i.e. $X=W_1+\cdots +W_r$. Then $X$ is distribued over Gamma:

$$\begin{array}{r}X\sim \Gamma (r,\lambda )\\ f_X(x)=\{\begin{array}{ll}\lambda e^{-\lambda x}\frac{{(\lambda x)}^{r-1}}{(r-1)!}& x>0\\ 1& \text{else}\end{array}\\ F_X(t)=1-\sum _{k=0}^{r-1}{e}^{-\lambda t}\frac{{(\lambda t)}^k}{k!}\\ \mathbb{E}(X)=\frac{r}{\lambda }\mathbb{SD}(X)=\frac{\sqrt{r}}{\lambda }\end{array}$$

Tip

Gamma Distribution also follows the Central Limit Theorem. Thus ${\lim }_{r\to \infty }X\sim \text{Normal}$