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Poisson Distribution

Statistical Theory and Methods

  1. Marc Hallin1,2

Published Online: 15 JAN 2013

DOI: 10.1002/9780470057339.vnn099

Encyclopedia of Environmetrics

Encyclopedia of Environmetrics

How to Cite

Hallin, M. 2013. Poisson Distribution. Encyclopedia of Environmetrics. 4.

Author Information

  1. 1

    ECARES, Université libre de Bruxelles, Bruxelles, Belgium

  2. 2

    ORFE, Princeton University, Princeton, NJ, USA

Publication History

  1. Published Online: 15 JAN 2013


The random variable X taking values 0,1,2,…,x,… with probabilities pλ(x) = e−λλx/x!, where inline image 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 [RIGHTWARDS ARROW] ∞ and p [RIGHTWARDS ARROW] 0 in such a way that np [RIGHTWARDS ARROW] λ, of the binomial distribution with exponent n and parameter p. The family of Poisson distributions indexed by inline image 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.


  • binomial variable;
  • count process;
  • poisson process;
  • exponential families;
  • monotone likelihood ratio