Poisson process (Poisson process, mainland translation Poisson process, Poisson process, Boissin process, Poisson process, Poisson process, Boussin process, etc. ) was named after the French mathematician Poisson (178 1- 1840). Poisson process is a random process, which is defined by the time when an event occurs. We say that the stochastic process N(t) is a time-homogeneous one-dimensional Poisson process if it meets the following conditions:

The number of events in two mutually exclusive (non-overlapping) intervals is an independent random variable.

The probability distribution of the number of events in the time interval is:

Where λ is a positive number and is a fixed parameter, which is usually called arrival rate or intensity. Therefore, if the number of events in a given time interval is given, the random variable presents Poisson distribution, and its parameters are.

More generally, Poisson process is to assign a random number of events to each bounded time interval or each bounded region in a certain space (for example, Euclidean plane or three-dimensional Euclidean space), so that

The number of events in one time interval or spatial region is independent from the number of events in another mutually exclusive (non-overlapping) time interval or spatial region.

The number of events in each time interval or spatial region is a random variable and follows Poisson distribution. (Technically, more accurately, every set of finite measures is given a random variable with Poisson distribution. )

Poisson process is one of the most famous processes in Levi's process. The homogeneous Poisson process is also an example of the homogeneous continuous-time Markov process. One-dimensional Poisson process with homogeneous time is a pure birth process and the simplest example of birth and death process.