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.
be a sequence of real numbers in
converges to a finite limit
Then Since this leaves Using Stirling's approximation, it can be written: Letting
so: 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.)