law of rare events
Basically, when the size of the population is very large and the occurrence of certain event is rare, where , the probability of is very small, the binomial random variable can be approximated by a Poisson random variable.
Sketch of Proof. Let . So
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 be the number of fatal accidents in a year from a population of 500 auto policies, then with and . and so
Using the binomial distribution, we have
|Title||law of rare events|
|Date of creation||2013-03-22 14:39:32|
|Last modified on||2013-03-22 14:39:32|
|Last modified by||CWoo (3771)|