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Poisson process

A Poisson process is an integer-valued nondecreasing stochastic process (such as a stochastic function of time N(t) as discussed here). Just as a Poisson random variable is characterized by its scalar parameter λ, a Poisson process is characterized by its rate function λ(t), which is the expected number of "events" or "arrivals" that occur per unit time. A homogeneous Poisson process has a constant parameter function λ(t) = λ and its marginal distribution N(a) has a Poisson distribution with parameter λa. In its most general form, the only two conditions for a (not necessarily homogeneous) Poisson process are:

  • Orderliness: which roughly means limΔ → 0 Pr[N(t+Δ)-N(t) > 1 | N(t+Δ)-N(t) ≥ 1] → 0 which implies that events don't occur simultaneously (but is actually a stronger statement).
  • Memorylessness (Also called Evolution without Aftereffects): Any event occurring after time t is independent of any event occurring before time t.

These seemingly unrestrictive conditions actually impose a great deal of structure in the Poisson process. In particular, they imply independent exponential (memoryless) interarrival times (with parameter λ for homogeneous processes). Because the interarrival times are exponentially distributed, the time between the 4th and 9th arrival (for instance) is distributed as the sum of exponential random variables (i.e. 5th order gamma distribution). Also, these conditions imply that the number of events in the interval [a,b), which is also written as N(b) - N(a) is Poisson distributed, (with parameter λ(b-a) for homogeneous processes).

This is a sample one-dimensional homogeneous Poisson process, N(t); not to be confused with a density or distribution function.


Sample Homogeneous Poisson Process

The Poisson-distributed random variables associated with different intervals are independent if and only if the intervals are disjoint. Each such Poisson-distributed random variable is said to count the number of "arrivals", "occurrences", "events" or "points" in the interval with which it is associated. (This makes the word "event" somewhat overworked, given its other uses in probability theory, and some prefer other terms on that account.)

Poisson processes can be generalized to multiple dimensions. A d-dimensional Poisson process associates with each region of finite volume in d-dimensional space a Poisson-distributed random variable with expected value r times the volume. Two or more such Poisson-distributed random variables are independent if the regions with which they are associated are disjoint or if their overlapping regions have rate function zero.



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