5.1 Poisson distribution for counts (discrete)

The Poisson distribution is a discrete distribution used for modelling counts.

Definition 1.1 (Poisson distribution) The Poisson distribution has the PMF \[ p_X(x) = \frac{\exp(-\mu)\mu^x}{x!}\quad\text{for $x=0, 1, 2, \dots$}, \] for \(\mu>0\).

A Poisson process is a process in which events occur continuously and independently at a constant average rate \(\mu\) according to a Poisson distribution.