# Poisson Statistics

In probability theory, a **Poisson process** is a stochastic process which counts the number of events^{[note 1]} and the time that these events occur in a given time interval. The time between each pair of consecutive events has an exponential distribution with parameter *λ* and each of these inter-arrival times is assumed to be independent of other inter-arrival times.

The Poisson process is a continuous-time process; the sum of a Bernoulli process can be thought of as its discrete-time counterpart. A Poisson process is a pure-birth process, the simplest example of a birth-death process. It is also a point process on the real half-line.

## Definition

The basic form of Poisson process, often referred to simply as "the Poisson process", is a continuous-time counting process {*N*(*t*), *t* ≥ 0} that possesses the following properties:

*N*(0) = 0- Independent increments (the numbers of occurrences counted in disjoint intervals are independent from each other)
- Stationary increments (the probability distribution of the number of occurrences counted in any time interval only depends on the length of the interval)
- The probability distribution of
*N*(*t*) is a Poisson distribution. - No counted occurrences are simultaneous.

Consequences of this definition include:

- The probability distribution of the waiting time until the next occurrence is an exponential distribution.
- The occurrences are Uniform distribution (continuous)|distributed uniformly on any interval of time. (Note that
*N*(*t*), the total number of occurrences, has a Poisson distribution over (0,*t*], whereas the location of an individual occurrence on*t*∈ (*a*,*b*] is uniform.)

Other types of Poisson process are described below.

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- ↑ The word
*event*used here is not an instance of the concept of event (probability theory)|*event*as frequently used in probability theory.