Poisson distribution

Typical Poisson distribution

In probability and statistics, Poisson distribution is a probability distribution. It is named after Siméon Denis Poisson. It measures the probability that a certain number of events occur within a certain period of time. The events need to be unrelated to each other. They also need to occur with a known average rate, represented by the symbol [math]\displaystyle{ \lambda }[/math] (lambda).

More specifically, if a random variable [math]\displaystyle{ X }[/math] follows Poisson distribution with rate [math]\displaystyle{ \lambda }[/math], then the probability of the different values of [math]\displaystyle{ X }[/math] can be described as follows:

[math]\displaystyle{ P(X=x)=\frac{e^{-\lambda} \lambda^x}{x!} }[/math] for [math]\displaystyle{ x = 0, 1, 2, \ldots }[/math]

Examples of Poisson distribution include:

• The numbers of cars that pass on a certain road in a certain time
• The number of telephone calls a call center receives per minute
• The number of light bulbs that burn out (fail) in a certain amount of time
• The number of mutations in a given stretch of DNA after a certain amount of radiation
• The number of errors that occur in a system
• The number of Property & Casualty insurance claims experienced in a given period of time

Poisson Distribution Media

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  Plot of the probability mass function for the Poisson distribution.

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  Plot of the cumulative distribution function for a Poisson distribution

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  Chewing gum on a sidewalk. The number of chewing gums on a single tile is approximately Poisson distributed.

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  Comparison of the Poisson distribution (black lines) and the binomial distribution with n = 10 (red circles), n = 20 (blue circles), n = 1000 (green circles). All distributions have a mean of 5.

Related pages

• Binomial distribution
• Exponential distribution