Poisson Distribution

probability-theory

Definition

Poisson Distribution

A Poisson distribution models the number of times an event occurs in a fixed interval when events happen independently and with a constant average rate.

Probability Mass Function

It can be modelled using the pmf:

$$p_k=P(\{k\})=e^{-\lambda }\cdot \frac{{\lambda }^k}{k!}$$

with $k\in {\mathbb{N}}_0$, and $\lambda \in (0,\infty )$ being the expected count per interval.

Expected Value

The expected value is given by:

$$\mathbb{E}[X]=\lambda$$

Variance

The variance is given by:

$$\mathbb{V}ar(X)=\lambda$$