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 ∈ (a, b] is uniform.)
Other types of Poisson process are described below.
Read more about this topic: Poisson Process
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