Poisson Distribution

The Poisson distribution, named after French mathematician Siméon Denis Poisson, is useful for counting relatively rare events, such as the number of insurance claims in a month, customer arrivals in an hour, defaults in a year or system failures in a week.

It is described by a single number: the average rate of events. For example, if an insurer receives an average of 3 large claims a year, the Poisson distribution gives the probability of receiving no large claims in a year as about 5%, and the probability of receiving 5 or more as about 18%.

Insurers use the Poisson distribution to model claim frequency. Banks use it for operational risk events. It is also used in queuing problems, such as staffing call centres, and in modelling jumps in asset prices.

The distribution assumes events are independent and occur at a constant rate. In practice, events often cluster, for example many claims after a storm, which can make the Poisson distribution underestimate the probability of extreme counts.