# Poisson Distribution

Pocket

By using $\lambda$ as a parameter, the Poisson distribution is defined as follows:

$f(k)=P(X=k)=\frac{\lambda^ke^{-\lambda}}{k!}$

The expectation value and variance of Poisson distribution can be computed as follows:

$E[X]=\int_{-\infty}^\infty kf(k) dk = \lambda$

$V[X]=\int_{-\infty}^\infty (k-E[X])^2f(k) dk = \lambda$

The Generating function of the Poisson distribution can be evaluated as follows:

$G[x]=\sum_{k=0}^\infty f(k)x^k=e^{\lambda(x-1)}$