Poisson point process

In probability theory, statistics 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 mathematical object that consists of points randomly located on a mathematical space 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 Poisson distribution. The process and the distribution are named after French mathematician Siméon Denis Poisson. The process itself was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and actuarial science. This point process is used as a mathematical model for seemingly random processes in numerous disciplines including astronomy, biology, ecology, geology, seismology, physics, economics, image processing, and telecommunications. The Poisson point process is often defined on the real number line, where it can be viewed as a stochastic process. It is used, for example, in queueing theory to model random events distributed in time, such as the arrival of customers at a store, phone calls at an telephone exchange, or the occurrence of earthquakes. In the plane, the point process—also known as a spatial Poisson process—can represent the locations of scattered objects such as transmitters in a wireless network, particles colliding into a particle detector, or trees in a forest. The process is widely used in mathematical models and in related fields, including spatial point processes, stochastic geometry, spatial statistics and continuum percolation theory. The point process depends on a single mathematical object, which, depending on the context, may be a constant, a locally integrable function or, in more general settings, a Radon measure. In the first case, the constant, known as the rate or intensity, is the average density of...