# Poisson distribution¶

## Story¶

The number $$n$$ of arrivals of a Poisson process in unit time is Poisson distributed.

## Example¶

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

## Parameters¶

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

## Support¶

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

## Probability mass function¶

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

## Moments¶

Mean: $$\lambda$$

Variance: $$\lambda$$

## Usage¶

Package

Syntax

NumPy

rg.poisson(lam)

SciPy

scipy.stats.poisson(lam)

Stan

poisson(lam)