\begin{definition}[Definition:Poisson Distribution]
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Then $X$ has the '''Poisson distribution with parameter $\lambda$''' (where $\lambda > 0$) {{iff}}:
:$\Img X = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \dfrac 1 {k!} \lambda^k e^{-\lambda}$
It is written:
:$X \sim \Poisson \lambda$
\end{definition}