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).


As and such that the mean value remains constant, we can approximate


Using Stirling's approximation, we can write:

Letting and :

As , so:

Alternative Proof[edit]

A simpler proof is possible without using Stirling's approximation:




This leaves


Ordinary Generating Functions[edit]

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, we find

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

See also[edit]


  1. ^ Papoulis, Pillai, Probability, Random Variables, and Stochastic Processes, 4th Edition