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one of the most important probability distributions of random variables that assume integral values. A random variable X that obeys a Poisson distribution takes on only nonnegative values; the probability that X = k is
where λ is a positive parameter. The distribution is named after S. D. Poisson; it first appeared in a work by him published in 1837. The mathematical expectation and variance of a random variable having a Poisson distribution with the parameter λ are equal to λ. If the independent random variables X1 and X2 have Poisson distributions with parameters λ1 and λ2, their sum X1 + X2 has a Poisson distribution with the parameter λ1 + λ2. Illustrations of Poisson distributions are given in Figure 1.
In probability-theoretic models, the Poisson distribution is used both as an approximating and an exact distribution. For example, if the events A1,…, An occur in n independent trials with the same low probability p, the probability that k of the n events simultaneously occur is approximated by the function pk(np); the mathematical significance of this assertion for large n and 1/p is stated by Poisson’s theorem. In particular, such a model is an excellent description of the process of radioactive decay and of numerous other physical phenomena.
The Poisson distribution is used as an exact distribution in the theory of stochastic processes. An example is the calculation of the load on communication lines. The number of calls made in nonoverlapping intervals of time are usually assumed to be independent random variables that obey a Poisson distribution with parameters whose values are proportional to the lengths of the corresponding intervals.
The arithmetic mean X̄ = (X1 + … + Xn)/n of n observed values of the random variables X1,…, Xn is used as an estimate of the unknown parameter λ, since this estimate is unbiased and has a minimum standard deviation.
REFERENCESGnedenko, B. V. Kurs teorii veroiatnostei, 5th ed. Moscow-Leningrad, 1969.
Feller, W. Vvedenie v teoriiu veroiatnostei i ee prilozheniia, 2nd ed., vol. 1. Moscow, 1967. (Translated from English.)
Poisson distributions are often used in building simulated user loads.