# Poisson distribution

discrete probability 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 ${\displaystyle \lambda }$ (lambda).[1]

More specifically, if a random variable ${\displaystyle X}$ follows Poisson distribution with rate ${\displaystyle \lambda }$, then the probability of the different values of ${\displaystyle X}$ can be described as follows:[2][3]

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

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

## References

1. "List of Probability and Statistics Symbols". Math Vault. 2020-04-26. Retrieved 2020-10-06.
2. "1.3.6.6.19. Poisson Distribution". www.itl.nist.gov. Retrieved 2020-10-06.
3. Weisstein, Eric W. "Poisson Distribution". mathworld.wolfram.com. Retrieved 2020-10-06.