# Poisson distribution

From AMS Glossary

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## Poisson distribution

A one-parameter, discrete frequency distribution giving the probability that

*n*points (or events) will be (or occur) in an interval (*or*time)*x*, provided that these points are individually independent and that the number occurring in a subinterval does not influence the number occurring in any other nonoverlapping subinterval.It has the form

*P*(*n*,*x*) =*e*^{-κx}(κ*x*)^{n}/*n*!. The mean and variance are both κ*x*, and κ is the average density (or rate) with which the events occur. When κ*x*is large, the Poisson distribution approaches the normal distribution. The binomial distribution approaches the Poisson when the number of events*n*becomes large and the probability of success*p*becomes small in such a way that*np*→ κ*x*. The Poisson distribution arises in such problems as radioactive and photoelectric emissions, thermal noise, service demands, and telephone traffic.