Poisson limit theorem

(Redirected from Poisson Theorem)

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.).