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In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson (//; French pronunciation: [pwasɔ̃]). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume. It plays an important role for discrete-stable distributions.
Probability mass function
Cumulative distribution function
|Support||(Natural numbers starting from 0)|
!}},} or or(for where is the upper incomplete gamma function, is the floor function, and is the regularized gamma function)
or for large
For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3. The most likely number of calls received are 2 and 3, but 1 and 4 are also likely. There is a small probability of it being as low as zero and a very small probability it could be 10 or even higher.
Another example is the number of decay events that occur from a radioactive source during a defined observation period.
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