Poisson limit theorem
- , such that
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
After simplifying the fraction:
After using the condition :
Apply, that due to we get :
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.
Proof using Ordinary Generating Functions
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 simply the OGF for the Poisson distribution (the second equality holds due to the definition of the Exponential function).
- Papoulis, Pillai, Probability, Random Variables, and Stochastic Processes, 4th Edition