# Poisson process

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A Poisson process, named after the French mathematician Siméon-Denis Poisson (1781–1840), is a stochastic process in which events occur continuously and independently of one another (the word event used here is not an instance of the concept of event frequently used in probability theory). Examples that are well-modeled as Poisson processes include the radioactive decay of atoms, telephone calls arriving at a switchboard, page view requests to a website, and rainfall.

The Poisson process is a collection {N(t) : t ≥ 0} of random variables, where N(t) is the number of events that have occurred up to time t (starting from time 0). The number of events between time a and time b is given as N(b) − N(a) and has a Poisson distribution. Each realization of the process {N(t)} is a non-negative integer-valued step function that is non-decreasing, but for intuitive purposes it is usually easier to think of it as a point pattern on [0,∞) (the points in time where the step function jumps, i.e. the points in time where an event occurs).

The Poisson process is a continuous-time process; the Bernoulli process can be thought of as its discrete-time counterpart (although strictly, one would need to sum the events in a Bernoulli process to also have a counting process). A Poisson process is a pure-birth process, the simplest example of a birth-death process. By the aforementioned interpretation as a random point pattern on [0, ∞) it is also a point process on the real half-line.

## Contents

### 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)
• No counted occurrences are simultaneous.

Consequences of this definition include:

• The probability distribution of N(t) is a Poisson distribution.
• The probability distribution of the waiting time until the next occurrence is an exponential distribution.
• The occurrences are 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 $t \in (a,b]$ is uniform.)