Poisson limit theorem

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

The law of rare events or Poisson limit theorem gives a Poisson approximation to the binomial distribution, under certain conditions.[1] The theorem was named after Siméon Denis Poisson (1781–1840).

Statement[edit]

If

, such that

then

Example[edit]

Suppose that in an interval [0, 1000], 500 points are placed randomly. Now what is the number of points that will be placed in [0, 10]?

The probabilistically precise way of describing the number of points in the sub-interval would be to describe it as a binomial distribution .

If we look here, the probability (that a random point will be placed in the sub-interval) is . Here so .

The probability that points lie in the sub-interval is

where: is the probability of falling with in the interval. gives the number of ways in which elements can be selected. gives the probability of the elements falling in the interval. counts the probability that elements fall outside of the interval

But using the Poisson Theorem we can approximate it as

Proofs[edit]

According to factorial's rate of growth, we replace factorials of large numbers with approximations:

After simplifying the fraction:

After using the condition :

As , so:

Q.E.D.

Alternative Proof[edit]

If we make the stronger assumption (rather than ) then a simpler proof is possible without needing approximations for the factorials. Since , we can rewrite . We now have:

Taking each of these terms in sequence, , meaning that .

Now . The first portion of this converges to , and the second portion goes to 1, as

This leaves us with . Q.E.D.

Ordinary Generating Functions[edit]

It is also possible to demonstrate the theorem through the use of Ordinary Generating Functions (OGF). Indeed, the OGF of the binomial distribution is

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 also[edit]

References[edit]

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