# Poisson distribution

M. Hallin

Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-review

### Abstract

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.
Original language English Encyclopedia of Environmetrics, 2nd Edition W. Piegorsch, A. El Shaarawi Wiley 1812-1814 3510 9780470973882 Published - 2012

### Fingerprint

Poisson distribution
Binomial distribution
Rare Events
Count Data
Exponential Family
Poisson process
Statistic
Siméon Denis Poisson
Random variable
Exponent
Sufficient
Modeling

### Cite this

Hallin, M. (2012). Poisson distribution. In W. Piegorsch, & A. El Shaarawi (Eds.), Encyclopedia of Environmetrics, 2nd Edition (pp. 1812-1814). Wiley.
Hallin, M. / Poisson distribution. Encyclopedia of Environmetrics, 2nd Edition. editor / W. Piegorsch ; A. El Shaarawi. Wiley, 2012. pp. 1812-1814
@inbook{6c21bbbcbb124d5b940bd3bc54272e08,
title = "Poisson distribution",
abstract = "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.",
author = "M. Hallin",
note = "Pagination: 3510",
year = "2012",
language = "English",
isbn = "9780470973882",
pages = "1812--1814",
editor = "W. Piegorsch and {El Shaarawi}, A.",
booktitle = "Encyclopedia of Environmetrics, 2nd Edition",
publisher = "Wiley",

}

Hallin, M 2012, Poisson distribution. in W Piegorsch & A El Shaarawi (eds), Encyclopedia of Environmetrics, 2nd Edition. Wiley, pp. 1812-1814.

Poisson distribution. / Hallin, M.

Encyclopedia of Environmetrics, 2nd Edition. ed. / W. Piegorsch; A. El Shaarawi. Wiley, 2012. p. 1812-1814.

Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-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 -

Hallin M. Poisson distribution. In Piegorsch W, El Shaarawi A, editors, Encyclopedia of Environmetrics, 2nd Edition. Wiley. 2012. p. 1812-1814