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Poisson Distribution (Pois₍λ₎) ○◂｜Definition｜1st｜20251119205401-00-⌔
Poisson distribution - Wikipedia
Poisson distribution
In probability theory and statistics, the Poisson distribution (/ˈpwɑːsɒn/) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event.1 It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1 (e.g., number of events in a given area or volume). The Poisson distribution is named after French mathematician Siméon Denis Poisson. It plays an important role for discrete-stable distributions.
Under a Poisson distribution with the expectation of λ events in a given interval, the probability of k events in the same interval is:2
For instance, consider a call center which receives an average of λ = 3 calls per minute at all times of day. If the number of calls received in any two given disjoint time intervals is independent, then the number k of calls received during any minute has a Poisson probability distribution. Receiving k = 1 to 4 calls then has a probability of about 0.77, while receiving 0 or at least 5 calls has a probability of about 0.23.
A classic example used to motivate the Poisson distribution is the number of radioactive decay events during a fixed observation period.3
Printed 2026-06-28.
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Link to original Footnotes
Haight, Frank A. (1967). Handbook of the Poisson Distribution. New York, NY, US: John Wiley & Sons. ISBN 978-0-471-33932-8. ↩
Yates, Roy D.; Goodman, David J. (2014). Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers (2nd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-45259-1. ↩
Ross, Sheldon M. (2014). Introduction to Probability Models (11th ed.). Academic Press. ↩
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