Poisson point process

Poisson Process

Probability density function
Mean	${\displaystyle a_{0,t}=\int _{0}^{t}\lambda (\alpha )d\alpha }$
Variance	${\displaystyle a_{0,t}+(a_{0,t})^{2}-(a_{0,t})^{2}=a_{0,t}}$ since ${\displaystyle R_{x}(t_{1},t_{2})=a_{0,min(t_{1},t_{2})}+a_{0,t_{1}}a_{0,t_{2}}}$ where for ${\displaystyle E\{X^{2}\}=R_{x}(t,t)=a_{0,t}+(a_{0,t})^{2}}$

In probability theory, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another.^[1] 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 real number line, it is often called simply the Poisson process.

This point process has convenient mathematical properties,^[2] which has led to its being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines including astronomy,^[3] biology,^[4] ecology,^[5]geology,^[6] seismology,^[7] physics,^[8] economics,^[9] image processing,^[10][11] and telecommunications.^[12][13]

The process's name derives from the fact that if a collection of random points form a Poisson process, then the number of points in a region of finite size is a random variable with 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.^[14][15]

The Poisson point process is often defined on the real number line, where it can be considered a stochastic process. It is used, for example, in queueing theory^[16] 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 plane, the point process, also known as a spatial Poisson process,^[17] can represent the locations of scattered objects such as transmitters in a wireless network,^{[12][18][19][20]} particles colliding into a detector or trees in a forest.^[21] The process is often used in mathematical models and in the related fields of spatial point processes,^[22] stochastic geometry,^[1] spatial statistics^[22][23] and continuum percolation theory.^[24]

The Poisson point process can be defined on more abstract spaces. Beyond applications, the Poisson point process is an object of mathematical study in its own right.^[2] 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.^[25] Modeling a system as a Poisson process is insufficient when the point-to-point interactions are too strong (that is, the points are not stochastically independent). Such a system may be better modeled with a different point process.^[26]

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.^[27] In the first case, the constant, known as the rate or intensity, is the average density 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.^[28] 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.^[29] The word point is often omitted,^[2] but there are other Poisson processes of objects, which, instead of points, consist of more complicated mathematical objects such as lines and polygons, and such processes can be based on the Poisson point process.^[30] Both the homogeneous and nonhomogeneous Poisson point processes are particular cases of the generalized renewal process.