Poisson Process - Definition

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.