### Abstract

*x*,… with probabilities pλ(x) = e

^{−λ}λ

^{x}/x!, where λ∈R

_{0}

^{+}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∈R

_{0}

^{+ }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.

Original language | English |
---|---|

Title of host publication | Encyclopedia of Environmetrics, 2nd Edition |

Editors | W. Piegorsch, A. El Shaarawi |

Publisher | Wiley |

Pages | 1812-1814 |

Number of pages | 3510 |

ISBN (Print) | 9780470973882 |

Publication status | Published - 2012 |

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### Cite this

*Encyclopedia of Environmetrics, 2nd Edition*(pp. 1812-1814). Wiley.

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*Encyclopedia of Environmetrics, 2nd Edition.*Wiley, pp. 1812-1814.

**Poisson distribution.** / Hallin, M.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Scientific › peer-review

TY - CHAP

T1 - Poisson distribution

AU - Hallin, M.

N1 - Pagination: 3510

PY - 2012

Y1 - 2012

N2 - The random variable X taking values 0,1,2,…,x,… with probabilities pλ(x) = e−λλx/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.

AB - The random variable X taking values 0,1,2,…,x,… with probabilities pλ(x) = e−λλx/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.

M3 - Chapter

SN - 9780470973882

SP - 1812

EP - 1814

BT - Encyclopedia of Environmetrics, 2nd Edition

A2 - Piegorsch, W.

A2 - El Shaarawi, A.

PB - Wiley

ER -