Poisson distribution

Story

The number \(n\) of arrivals of a Poisson process in unit time is Poisson distributed.

The number of mutations in a strand of DNA per unit length (since mutations are rare) are Poisson distributed.

The single parameter is the rate \(\lambda\) of the arrival of the Poisson process.

The Poisson distribution is supported on the set of nonnegative integers.

\[\begin{align} f(n;\lambda) = \frac{\lambda^n}{n!}\,\mathrm{e}^{-\lambda}. \end{align}\]

The CDF evaluated at nonnegative integer \(n\) is

\[\begin{align} F(n;\lambda) = \frac{\Gamma(n + 1, \lambda)}{n!}, \end{align}\]

where \(\Gamma(n + 1, \lambda)\) is the upper incomplete gamma function.

Mean: \(\lambda\)

Variance: \(\lambda\)

Consider \(K\) Poisson processes with arrival rates \(\lambda_1\), \(\lambda_2\), …, \(\lambda_K\). Let

\[\begin{align} \lambda = \sum_{i=1}^K \lambda_i. \end{align}\]

If \(N\) is the total number of arrivals of these \(K\) Poisson processes, \(N \sim \text{Poisson}(\lambda)\).