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 (1781–1840). 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:
First proof Assume (the case is easier). Then
Since
this leaves
Alternative proof Using Stirling's approximation, it can be written:
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, 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 De Moivre–Laplace theorem Le Cam's theorem
References