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(ba) for homogeneous processes).
This is a sample onedimensional homogeneous Poisson process, N(t); not to be confused with a density or distribution function.
The Poissondistributed random variables associated with different intervals are independent if and only if the intervals are disjoint. Each such Poissondistributed 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 ddimensional Poisson process associates with each region of finite volume in ddimensional space a Poissondistributed random variable with expected value r times the volume. Two or more such Poissondistributed 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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