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since <math>R_x(t_1,t_2) = a_{0, min(t_1,t_2)} + a_{0, t_1} a_{0, t_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]]
In [[probability theory]], [[statistics]] and related fields, a '''Poisson point process''' is a type of [[Randomness|random]] [[mathematical object]] that consists of [[Point (geometry)|points]] randomly located on a [[Space (mathematics)|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''' and '''Poisson point field'''. When the process is defined on the [[Number line|real number line]], it is often called simply the '''Poisson process.'''
This [[point process]] has convenient mathematical properties,<ref name="Kingman1992">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|date=17 December 1992|publisher=Clarendon Press|isbn=978-0-19-159124-2}}</ref> which has led to its being frequently defined in [[Euclidean space]] and used as a [[mathematical model]] for seemingly random processes in numerous disciplines
The process
The Poisson point process is often defined on the [[real line]], where it can be considered as a [[stochastic process]]. In this setting, it is used, for example, in [[queueing theory]]<ref name="Kleinrock1976">{{cite book|author=Leonard Kleinrock|title=Queueing Systems: Theory|url=https://archive.org/details/queueingsystems01klei|url-access=registration|year=1976|publisher=Wiley|isbn=978-0-471-49110-1}}</ref> to model random events, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes, distributed in time. In the [[Plane (geometry)|plane]], the point process, also known as a '''spatial Poisson process''',<ref name="BaddeleyBárány2006page10">{{cite book|author1=A. Baddeley|author2=I. Bárány|author3=R. Schneider|title=Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004|url=https://books.google.com/books?id=X-m5BQAAQBAJ|date=26 October 2006|publisher=Springer|isbn=978-3-540-38175-4|page=10}}</ref> can represent the locations of scattered objects such as transmitters in a [[wireless network]],<ref name="baccelli2009stochastic2"/><ref name="andrews2010primer">J. G. Andrews, R. K. Ganti, M. Haenggi, N. Jindal, and S. Weber. A primer on spatial modeling and analysis in wireless networks. ''Communications Magazine, IEEE'', 48(11):156–163, 2010.</ref><ref name="baccelli2009stochastic1">F. Baccelli and B. Błaszczyszyn. ''Stochastic Geometry and Wireless Networks, Volume I – Theory'', volume 3, No 3–4 of ''Foundations and Trends in Networking''. NoW Publishers, 2009.</ref><ref name="Haenggi2013">{{cite book|author=Martin Haenggi|title=Stochastic Geometry for Wireless Networks|url=https://books.google.com/books?id=CLtDhblwWEgC|year=2013|publisher=Cambridge University Press|isbn=978-1-107-01469-5}}</ref> [[particles]] colliding into a detector, or trees in a forest.<ref name="ChiuStoyan2013page51"/> In this setting, the process is often used in mathematical models and in the related fields of spatial point processes,<ref name="BaddeleyBárány2006">{{cite book|author1=A. Baddeley|author2=I. Bárány|author3=R. Schneider|title=Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004|url=https://books.google.com/books?id=X-m5BQAAQBAJ|date=26 October 2006|publisher=Springer|isbn=978-3-540-38175-4}}</ref> [[stochastic geometry]],<ref name="ChiuStoyan2013"/> [[spatial statistics]]<ref name="BaddeleyBárány2006"/><ref name="MollerWaagepetersen2003">{{cite book|author1=Jesper Moller|author2=Rasmus Plenge Waagepetersen|title=Statistical Inference and Simulation for Spatial Point Processes|url=https://books.google.com/books?id=dBNOHvElXZ4C|date=25 September 2003|publisher=CRC Press|isbn=978-0-203-49693-0}}</ref> and [[continuum percolation theory]].<ref name="meester1996continuum">R. Meester and R. Roy. Continuum percolation, volume 119 of cambridge tracts in mathematics, 1996.</ref> The Poisson point process can be defined on more [[Abstraction (mathematics)|abstract]] spaces. Beyond applications, the Poisson point process is an object of mathematical study in its own right.<ref name="Kingman1992"/> In all settings, the Poisson point process has the property that each point is [[stochastically independent]] to all the other points in the process, which is why it is sometimes called a ''purely'' or ''completely'' random process.{{sfnp|Daley|Vere-Jones|2003|page=27}} Modeling a system as a Poisson Process is insufficient when the point-to-point interactions are too strong (i.e. the points are not stochastically independent). Such a system may be better modeled with a different point process.<ref name="ChiuStoyan2013page35">{{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|pages=35–36}}</ref>
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