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A Poisson process, named after the French mathematician SiméonDenis 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 wellmodeled 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 nonnegative integervalued step function that is nondecreasing, 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 continuoustime process; the Bernoulli process can be thought of as its discretetime 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 purebirth process, the simplest example of a birthdeath process. By the aforementioned interpretation as a random point pattern on [0, ∞) it is also a point process on the real halfline.
Contents
Definition
The basic form of Poisson process, often referred to simply as "the Poisson process", is a continuoustime 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:
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