In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

For instance, an individual keeping track of the amount of mail they receive each day may notice that they receive an average number of 4 letters per day. As it is reasonable to assume that receiving one piece of mail will not affect the arrival times of future pieces of mail—that pieces of mail from a wide range of sources arrive independently of one another—the number of pieces of mail received per day would obey a Poisson distribution. Other examples might include: the number of phone calls received by a call center per hour, the number of decay events per second from a radioactive source, or the number of taxis passing a particular street corner per hour.

## Definition

A discrete random variable is said to have a Poisson distribution with parameter λ > 0, if, for k = 0, 1, 2, …, the probability mass function of is given by: $\!f(k; \lambda)= \Pr(X=k)= \frac{\lambda^k e^{-\lambda}}{k!},$

where

The positive real number λ is equal to the expected value of X and also to its variance $\lambda=\operatorname{E}(X)=\operatorname{Var}(X).$

The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. How many such events will occur during a fixed time interval? Under the right circumstances, this is a random number with a Poisson distribution.

## Occurrence

Applications of the Poisson distribution can be found in many fields related to counting:

The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete properties (that is, those that may happen 0, 1, 2, 3, ... times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or space. Examples of events that may be modelled as a Poisson distribution include:

• The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry. This example was made famous by a book of Ladislaus Josephovich Bortkiewicz (1868–1931).
• The number of yeast cells used when brewing Guinness beer. This example was made famous by William Sealy Gosset (1876–1937).[dead link]
• The number of phone calls arriving at a call centre within a minute. This example was made famous by A.K. Erlang (1878 – 1929).
• Internet traffic.
• The number of goals in sports involving two competing teams.
• The number of deaths per year in a given age group.
• The number of jumps in a stock price in a given time interval.
• Under an assumption of homogeneity, the number of times a web server is accessed per minute.
• The number of mutations in a given stretch of DNA after a certain amount of radiation.
• The proportion of cells that will be infected at a given multiplicity of infection.
• The arrival of photons on a pixel circuit at a given illumination and over a given time period.
• The targeting of V-1 flying bombs on London during World War II.

Gallagher in 1976 showed that the counts of prime numbers in short intervals obey a Poisson distribution provided a certain version of an unproved conjecture of Hardy and Littlewood is true.