# 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.

Title Poisson random variable PoissonRandomVariable 2013-03-22 11:54:03 2013-03-22 11:54:03 Koro (127) Koro (127) 13 Koro (127) Definition msc 62E15 msc 92B05 msc 92-01 Poisson distribution