26 Facts About Poisson process

In probability, 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.

The Poisson process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics.

Poisson point process is often defined on the real line, where it can be considered as a stochastic process.

Beyond applications, the Poisson point process is an object of mathematical study in its own right.

The Poisson process distribution is the probability distribution of a random variable such that the probability that equals is given by:.

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In other words, there is a lack of interaction between different regions and the points in general, which motivates the Poisson process being sometimes called a purely or completely random process.

The parameter, called rate or intensity, is related to the expected number of Poisson process points existing in some bounded region, where rate is usually used when the underlying space has one dimension.

Poisson counting process can be defined by stating that the time differences between events of the counting process are exponential variables with mean.

In that case the Poisson process is no longer stationary, according to some definitions of stationarity.

Spatial Poisson process is a Poisson point process defined in the plane.

Similarly to the one-dimensional case, the homogeneous point Poisson process is restricted to some bounded subset of, then depending on some definitions of stationarity, the Poisson process is no longer stationary.

An inhomogeneous Poisson process defined in the plane is called a spatial Poisson process It is defined with intensity function and its intensity measure is obtained performing a surface integral of its intensity function over some region.

The intensity measure of this point Poisson process is dependent on the location of underlying space, which means it can be used to model phenomena with a density that varies over some region.

Poisson point process can be further generalized to what is sometimes known as the general Poisson point process or general Poisson process by using a Radon measure, which is locally-finite measure.

The name stems from its inherent relation to the Poisson process distribution, derived by Poisson process as a limiting case of the binomial distribution.

Poisson process derived the Poisson process distribution, published in 1841, by examining the binomial distribution in the limit of and .

For example, John Michell in 1767, a decade before Poisson process was born, was interested in the probability a star being within a certain region of another star under the assumption that the stars were "scattered by mere chance", and studied an example consisting of the six brightest stars in the Pleiades, without deriving the Poisson process distribution.

Poisson process then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.

For example, for a point Poisson process defined on the Euclidean state space and a function on, the expression.

Avoidance function or void probability of a point Poisson process is defined in relation to some set, which is a subset of the underlying space, as the probability of no points of existing in.

One example of an operation is known as thinning which entails deleting or removing the points of some point Poisson process according to a rule, creating a new Poisson process with the remaining points .

Displacement theorem can be extended such that the Poisson process points are randomly displaced from one Euclidean space to another Euclidean space, where is not necessarily equal to.

Tractability of the Poisson process means that sometimes it is convenient to approximate a non-Poisson point process with a Poisson one.

Poisson point process can be generalized by, for example, changing its intensity measure or defining on more general mathematical spaces.

Cox point process, Cox process or doubly stochastic Poisson process is a generalization of the Poisson point process by letting its intensity measure to be random and independent of the underlying Poisson process.

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Compound Poisson point process or compound Poisson process is formed by adding random values or weights to each point of Poisson point process defined on some underlying space, so the process is constructed from a marked Poisson point process, where the marks form a collection of independent and identically distributed non-negative random variables.