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Probability Theory
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.
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
References