"Poisson theorem" redirects here. For the "Poisson's theorem" in Hamiltonian mechanics, see Poisson bracket § Constants of motion.

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).

Contents

TheoremEdit

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

 

ProofsEdit

Using Stirling's approximation, we can write:

 

Letting   and  :

 

As  ,   so:

 

Alternative ProofEdit

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

 .

Since

 

and

 

This leaves

 .

Ordinary Generating FunctionsEdit

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 alsoEdit

ReferencesEdit

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