1. 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.

2. Poisson Probability Distribution 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: 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.

3. 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:

4. 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