# 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 $\lambda$ (lambda).

More specifically, if a random variable $X$ follows Poisson distribution with rate $\lambda$ , then the probability of the different values of $X$ can be described as follows:

$P(X=x)={\frac {e^{-\lambda }\lambda ^{x}}{x!}}$ for $x=0,1,2,\ldots$ 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