Poisson distribution

A Poisson random variable is a discrete random variable which models the probability of occurrence for some event with a fixed rate of occurrence, independent of previous occurrence. Such a random variable follows a Poisson distribution, denoted $X\sim \mathrm{Pois}(\lambda )$, where $\lambda$ is the number of events which will occur in one unit of time.

Formally,
if $X$ is a Poisson random variable, i.e. $X\sim \mathrm{Pois}(\lambda )$, then the probability of $k$ occurrences in one unit of time is given by the probability mass function

where $e$ is the base of the natural logarithm.