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Poisson Theorem

Poisson's theorem states that if n->infty, p->0, and np->lambda, where lambda is fixed, then

(n!)/(k!(n-k)!)p^k(1-p)^(n-k)->e^(-lambda)(lambda^k)/(k!)

for each fixed nonnegative integer k. Thus, for large n and small p, the probability that an event occurs k times in n independent trials is approximated by

(n!)/(k!(n-k)!)p^kq^(n-k)∼e^(-np)((np)^k)/(k!),

where q=1-p. This is the Poisson distribution approximation to the binomial distribution, with mean lambda=np.

See also

Binomial Distribution, Le Cam's Inequality, Poisson Distribution

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References

Barbour, A. D.; Holst, L.; and Janson, S. Poisson Approximation. Oxford, England: Clarendon Press, 1992.Feller, W. "The Poisson Approximation." §6.5 in An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, pp. 153-155, 1968.Le Cam, L. "An Approximation Theorem for the Poisson Binomial Distribution." Pacific J. Math. 10, 1181-1197, 1960.Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 71, 1984.

Referenced on Wolfram|Alpha

Poisson Theorem

Cite this as:

Weisstein, Eric W. "Poisson Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PoissonTheorem.html

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