Poisson distribution

The random variable X taking values 0,1,2,…,x,… with probabilities pλ(x) = e^−λλ^x/x!, where λ∈R₀⁺ is called a Poisson variable, and its distribution a Poisson distribution, with parameter λ. The Poisson distribution with parameter λ can be obtained as the limit, as n → ∞ and p → 0 in such a way that np → λ, of the binomial distribution with exponent n and parameter p. The family of Poisson distributions indexed by λ∈R₀⁺ is an exponential family, with natural parameter logλ and privileged sufficient and complete statistic X. Poisson distributions are often used in the modeling of count data for “rare events.” As such, they also play a fundamental role in the so-called Poisson processes.