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 languageEnglish
    Title of host publicationEncyclopedia of Environmetrics, 2nd Edition
    EditorsW. Piegorsch, A. El Shaarawi
    PublisherWiley
    Pages1812-1814
    Number of pages3510
    ISBN (Print)9780470973882
    Publication statusPublished - 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
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    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.",
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    language = "English",
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    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

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    PY - 2012

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    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.

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    BT - Encyclopedia of Environmetrics, 2nd Edition

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