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| In [[probability theory]], a '''Poisson point process''' is a particular kind of [[stochastic process|random process]] by which a set of isolated points are scattered about a line or a plane or a three-dimensional space or any of various other sorts of spaces. The concept is named (perhaps erroneously) after the French mathematician [[Siméon Denis Poisson]]. Often the term [[Poisson process]] is used to mean a Poisson point process in which the space in which isolated points are randomly scattered is a line, which in many applications represents time.
| | Art Teacher (Private Tuition ) Vernon from La Malbaie, usually spends time with pastimes such as models, property developers in [http://voice4abetterindia.com/activity/p/50124/ singapore apartment for sale] and stamp collecting. Finds the beauty in traveling to spots around the entire world, recently only coming back from Generalife and Albayzín. |
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| The Poisson point process is [[characterization (mathematics)|characterized]] by the following properties:
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| * The numbers of isolated points falling within two regions ''A'' and ''B'' are [[independence (probability theory)|independent]] [[random variable]]s if ''A'' and ''B'' do not intersect each other;
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| * The [[expected value|expected]] number of isolated points falling within a region ''A'' is the measure of the region ''A''. This "measure" is often proportional to the area or volume of ''A'', but sometimes more elaborate measures are used. But the measure must be defined in such a way that the measure of the union of regions that do not intersect each other is simply the sum of their measures.
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| A consequence of these characterizing properties is that the probability distribution of the number ''X'' of isolated points falling within any region ''A'' is a [[Poisson distribution]], which means that
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| : <math> \Pr(X=x) = \frac{(\mu(A))^x e^{-\mu(A)}}{x!} </math>
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| where ''μ''(''A'') is the measure of the region ''A''.
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| ==Further reading==
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| *{{cite doi|10.1007/978-1-4419-6923-1}}
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| {{math-stub}}
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| {{Stochastic processes}}
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| [[Category:Stochastic processes]]
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| [[Category:Poisson processes]]
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| [[Category:Point processes]]
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| [[Category:Spatial processes]]
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Revision as of 21:44, 22 February 2014
Art Teacher (Private Tuition ) Vernon from La Malbaie, usually spends time with pastimes such as models, property developers in singapore apartment for sale and stamp collecting. Finds the beauty in traveling to spots around the entire world, recently only coming back from Generalife and Albayzín.