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Depending on the setting, the process has several equivalent definitions<ref name="Tijms2003page1">{{cite book|author=H. C. Tijms|title=A First Course in Stochastic Models|url=https://books.google.com/books?id=RK9yFrNxom8C|date=18 April 2003|publisher=John Wiley & Sons|isbn=978-0-471-49880-3|pages=1–2}}</ref> as well as definitions of varying generality owing to its many applications and characterizations.{{sfnp|Daley|Vere-Jones|2003|pages=26–37}} The Poisson point process can be defined, studied and used in one dimension, for example, on the real line, where it can be interpreted as a counting process or part of a queueing model;<ref name="Tijms2003page1and9">{{cite book|author=H. C. Tijms|title=A First Course in Stochastic Models|url=https://books.google.com/books?id=RK9yFrNxom8C|date=18 April 2003|publisher=John Wiley & Sons|isbn=978-0-471-49880-3|pages=1 and 9}}</ref><ref name="Ross1996page59">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|pages=59–60}}</ref> in higher dimensions such as the plane where it plays a role in [[stochastic geometry]]<ref name="ChiuStoyan2013"/> and [[spatial statistics]];<ref name="baddeley1999crash">A. Baddeley. A crash course in stochastic geometry. ''Stochastic Geometry: Likelihood and Computation Eds OE Barndorff-Nielsen, WS Kendall, HNN van Lieshout (London: Chapman and Hall)'', pages 1–35, 1999.</ref> or on more general mathematical spaces.<ref name="DaleyVere-Jones2007page1">{{cite book|author1=D.J. Daley|author2=David Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure|url=https://books.google.com/books?id=nPENXKw5kwcC|date=12 November 2007|publisher=Springer Science & Business Media|isbn=978-0-387-21337-8|pages=1–2}}</ref> Consequently, the notation, terminology and level of mathematical rigour used to define and study the Poisson point process and points processes in general vary according to the context.<ref name="ChiuStoyan2013page110to111">{{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=110–111 }}</ref>
Despite all this, the Poisson point process has two key properties—the Poisson property and the independence property— that play an essential role in all settings where the Poisson point process is used.<ref name="ChiuStoyan2013page41and51"/><ref name="Kingman1992page11"/> The two properties are not logically independent; indeed,
===Poisson distribution of point counts===
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