Poisson random variable

The Poisson discrete probability function with parameter $\lambda>0$ is given by

$f_{X}(x)=\frac{e^{-\lambda}\lambda^{x}}{x!},\quad\quad x\in\mathbb{N}.$

A random variable $X$ with such a density has expectation, variance, moment generating function and characteristic function given by $E[X]=\lambda$ , $Var[X]=\lambda$ , $M_{X}(t)=e^{\lambda(e^{t}-1)}$ , and $\phi_{X}(t)=e^{\lambda(e^{it}-1)}$ , respectively.