The Poisson Distribution

The Poisson Distribution

The Poisson distribution models the number of events that occur in a fixed interval of time or space when those events happen independently at a constant average rate.

The single parameter

A Poisson is defined by one parameter, lambda, the expected number of events per interval. A striking property is that its mean and variance are both equal to lambda. This makes the distribution easy to specify but also a useful check: if observed counts have variance far above the mean, the data is overdispersed and Poisson may be a poor fit.

When it applies

The classic conditions are events that are independent, occur at a steady rate, and do not happen simultaneously. Examples include:

• The number of emails arriving per hour.
• The number of typos on a page.
• The number of customers reaching a server per minute.

Link to the binomial

The Poisson is the limit of a binomial when the number of trials grows large and the success probability shrinks, with their product fixed at lambda. This is why it models many trials with a rare chance each.

Key idea

The Poisson distribution counts independent events at a constant rate over an interval, defined by lambda which equals both its mean and variance.