The Poisson Probability Distribution The Poisson probability distribution provides a good model for the probability distribution of the number of “rare events” that occur randomly in time, distance, or space.

Poisson Probability Distribution Assume that an interval is divided into a very large number of subintervals so that the probability of the occurrence of an event in any subinterval is very small. Assumptions of a Poisson probability distribution: distribution • The probability of an occurrence of an event is constant for all subintervals: independent events; • You are counting the number times a particular event occurs in a unit; and • As the unit gets smaller, the probability that two or more events will occur in that unit approaches zero.

Poisson Probability Distribution The random variable X is said to follow the Poisson probability distribution if it has the probability function: where • P(x) = the probability of x successes over a given period of time or space, given • = the expected number of successes per time or space unit; > 0 • e = 2. 71828 (the base for natural logarithms) • The mean and variance of the Poisson probability distribution are:

Example A life insurance company insures the lives of 5, 000 men of age 42. If actuarial studies show the probability of any 42 -year-old man dying in a given year to be 0. 001, the probability that the company will have to pay 4 claims in a given year can be approximated by the Poisson distribution. P ( X = 4 n = 5000, p = 0. 001 ) = 0. 1745