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.