Poisson point process

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, <a href="/facts/Statistics/DNPsKGYU">statistics</a> and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of <a href="/facts/Mathematical_object/Juhyow9M">mathematical object</a> that consists of <a href="/facts/Point_(geometry)/6YPAvX2a">points</a> randomly located on a <a href="/facts/Space_(mathematics)/G6rt4bQ8">mathematical space</a> with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson distribution</a>. The process and the distribution are named after French mathematician <a href="/facts/Sim%25C3%25A9on_Denis_Poisson/zmsDITrY">Siméon Denis Poisson</a>. The process itself was discovered independently and repeatedly in several settings, including experiments on <a href="/facts/Radioactive_decay/Ya6FVjt4">radioactive decay</a>, telephone call arrivals and <a href="/facts/Actuarial_science/KxmCE0ME">actuarial science</a>.
This point process is used as a <a href="/facts/Mathematical_model/CGhyuxgz">mathematical model</a> for seemingly random processes in numerous disciplines including <a href="/facts/Astronomy/0ArJVM1p">astronomy</a>, <a href="/facts/Biology/1M1ee3pB">biology</a>, <a href="/facts/Ecology/YE8upuJU">ecology</a>, <a href="/facts/Geology/WGvVRIYF">geology</a>, <a href="/facts/Seismology/Lony1LYX">seismology</a>, <a href="/facts/Physics/a7aH2y08">physics</a>, <a href="/facts/Economics/HYctA2Dn">economics</a>, <a href="/facts/Image_processing/zenalNCf">image processing</a>, and <a href="/facts/Telecommunications/vxoj1cxR">telecommunications</a>.
The Poisson point process is often defined on the real number line, where it can be considered a <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic process</a>. It is used, for example, in <a href="/facts/Queueing_theory/8NDl56EM">queueing theory</a> to model random events distributed in time, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes. In the <a href="/facts/Plane_(geometry)/tAUtRFcn">plane</a>, the point process, also known as a spatial Poisson process, can represent the locations of scattered objects such as transmitters in a <a href="/facts/Wireless_network/yr4SWTdl">wireless network</a>, <a href="/facts/Particles/JHKr2w6h">particles</a> colliding into a detector or trees in a forest. The process is often used in mathematical models and in the related fields of spatial point processes, <a href="/facts/Stochastic_geometry/r7BW9A8k">stochastic geometry</a>, <a href="/facts/Spatial_statistics/R6rlDzRl">spatial statistics</a> and <a href="/facts/Continuum_percolation_theory/c61b7IeB">continuum percolation theory</a>.
The point process depends on a single mathematical object, which, depending on the context, may be a <a href="/facts/Constant_(mathematics)/SxlxTWFT">constant</a>, a <a href="/facts/Locally_integrable_function/HgcUMNG6">locally integrable function</a> or, in more general settings, a <a href="/facts/Radon_measure/21adq3XN">Radon measure</a>. In the first case, the constant, known as the rate or intensity, is the average <a href="/facts/Density/92p6hh9I">density</a> of the points in the Poisson process located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process. In the second case, the point process is called an inhomogeneous or nonhomogeneous Poisson point process, and the average density of points depend on the location of the underlying space of the Poisson point process. The word point is often omitted, but there are other Poisson processes of objects, which, instead of points, consist of more complicated mathematical objects such as <a href="/facts/Line_(geometry)/gJqsr4Gj">lines</a> and <a href="/facts/Polygon/xvkmYhvG">polygons</a>, and such processes can be based on the Poisson point process. Both the homogeneous and nonhomogeneous Poisson point processes are particular cases of the <a href="/facts/Generalized_renewal_process/IXgJsNGM">generalized renewal process</a>.

Poisson point process open-in-new

Poisson point process