law of rare events

Let X be distributed as Bin(n,p), a binomial random variableMathworldPlanetmath with parameters n and p. Suppose


where λ is a positive real constant, then X is asymptotically distributed as Poisson(λ), a Poisson distributionMathworldPlanetmath with parameter λ.

Basically, when the size of the population n is very large and the occurrence of certain event A is rare, where p, the probability of A is very small, the binomial random variable X can be approximated by a Poisson random variable.

Sketch of Proof. Let XBin(n,p). So

P(X=m) = n!m!(n-m)!pm(1-p)n-m
= n!nm(n-m)!(np)mm!(1-npn)n-m
= n!nm(n-m)!(np)mm!(1-npn)n(1-npn)-m.

As n,






Example. Suppose in a given year, the number of fatal automobile accidents has a binomial distribution for a particular insuarance company with five hundred automobile insurance policies. On average, there is one policy out of the five hundred that will be involved in a fatal crash. What is the probability that there will be no fatal accidents (out of five hundred policies) in any particular year?

Solution. If X be the number of fatal accidents in a year from a population of 500 auto policies, then XBin(n,p) with n=500 and p=1/500. λ=500×1/500=1 and so


Using the binomial distribution, we have

Title law of rare events
Canonical name LawOfRareEvents
Date of creation 2013-03-22 14:39:32
Last modified on 2013-03-22 14:39:32
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 62P05
Classification msc 60E99
Classification msc 60F99
Synonym Poisson theorem