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