## Proof That the Poisson Distribution is a Probablity Distribution

In order for a function to be a probability distribution, it must sum or integrate over the possible values of the random variable(s) to 1.
A random variable
$X$
is said to be modelled by a Poisson distribution
$Po( \lambda)$
if
$X \sim \frac{e^{- \lambda} \lambda^x} {x!}$

\begin{aligned} \sum_{x=0}^{\infty} \frac{e^{- \lambda} \lambda^x} {x!} &= e^{- \lambda} \sum_{x=0}^{\infty} \frac{ \lambda^x } {x!} \\ &= e^{- \lambda} e^{ \lambda} =1 \end{aligned}