Poission Distribution

Definition

$X\sim \mathrm{Pois}(\lambda )$ where $\lambda$ is the average number of occurrences in a fixed interval of time

The number of occurrences in a fixed interval of time with mean $\lambda$

Properties

PDF

$f(x)=\frac{e^{-\lambda }{\lambda }^x}{x!}\in [0,1]$ where $x\in {\mathbb{N}}_0$

Mean

$E(X)=\lambda$

Variance

$\mathrm{Var}(X)=\lambda$

MGF

$M(t)=\exp [m(e^t-1)]$

Summation

Let $X_i\sim \mathrm{Pois}({\lambda }_i)$, and $X_i$‘s are independent, then $\sum _{i=1}^n{X}_i\sim \mathrm{Pois}\left(\sum _{i=1}^n{\lambda }_i\right)$