Poisson limit 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. The theorem was named after Siméon Denis Poisson. A generalization of this theorem is Le Cam's theorem.

Theorem

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

Proofs

Since
and
This leaves

Alternative proof

Using Stirling's approximation, we can write:
Letting and :
As, so:

Ordinary generating functions

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.