A Poisson process is a random function U(t) which describes the number of random events in an interval [0,t] of time or space.
The random events have the properties that (i) the probability of an event during a very small interval from t to t + h is rh, (ii) the probability of more than one event during such a time interval is negligible, (iii) the probability of an event during such a time interval does not depend on what happened prior to time t. The constant r here is the rate of events – that is, the average number of events on a unit interval.
The distribution of a Poisson process at time t is the Poisson distribution with its parameter l equal to rt.
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