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Poisson limit theorem

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In probability theory, the law of rare events or Poisson limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions.[1] The theorem was named after Siméon Denis Poisson (1781–1840). A generalization of this theorem is Le Cam's theorem.

Theorem

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Let be a sequence of real numbers in such that the sequence converges to a finite limit . Then:

First proof

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Assume (the case is easier). Then

Since

this leaves

Alternative proof

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Using Stirling's approximation, it can be written:

Letting and :

As , so:

Ordinary generating functions

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It is also possible to demonstrate the theorem through the use of ordinary generating functions of the binomial distribution:

by virtue of the binomial theorem. Taking the limit while keeping the product constant, it can be seen:

which is the OGF for the Poisson distribution. (The second equality holds due to the definition of the exponential function.)

See also

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References

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  1. ^ Papoulis, Athanasios; Pillai, S. Unnikrishna. Probability, Random Variables, and Stochastic Processes (4th ed.).