Poisson point process: Difference between revisions

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{{Short description|Type of random mathematical object}}

{{Use dmy dates|date=July 2021}}{{Infobox probability distribution|name=Poisson Process|pdf_image=[[File:Poisson Process.png|325px]]|mean=<math>a_0 t</math>|variance=<math>a_0 t + a_0 t^2 - (a_0 t )^2 </math>

{{Use dmy dates|date=July 2021}}

(since <math>R_x(t_1,t_2) = a_0 min(t_1,t_2) + a_0 t_1t_2</math> where for <math>E\{X^2\} = R_x(t,t) = a_0 t + a_0 t^2</math> )|type=multivariate}}[[File:Poisson process.svg|thumb|alt=Poisson point process|A visual depiction of a Poisson point process starting from 0, in which increments occur continuously and independently at rate ''λ''.]]

In [[probability]], statistics and related fields, a '''Poisson point process''' is a type of [[random]] [[mathematical object]] that consists of [[Point (geometry)|points]] randomly located on a [[mathematical space]] with the essential feature that the points occur independently of one another.<ref name="ChiuStoyan2013">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3}}</ref> The Poisson point process is also called a '''Poisson random measure''', '''Poisson random point field''' or '''Poisson point field'''. When the process is defined on the [[real line]], it is often called simply the '''Poisson process.'''