The Poisson Distribution. Another important discrete probability distribution is the Poisson distribution, named in honor of the French mathematician and physicist Simeon Poisson (1781–1840).

This probability distribution is often used to model the frequency with which a specified event occurs during a particular period of time. The Poisson probability formula is P(X = x) = e−λ λx x!

, where X is the number of times the event occurs and λ is a parameter equal to the mean of X. The number e is the base of natural logarithms and is approximately equal to 2.7183.

To illustrate, consider the following problem: Desert Samaritan Hospital, located in Mesa, Arizona, keeps records of emergency room traffic. Those records reveal that the number of patients who arrive between 6:00 P.M. and 7:00 P.M. has a Poisson distribution with parameter λ = 6.9. Determine the probability that, on a given day, the number of patients who arrive at the emergency room between 6:00 P.M. and 7:00 P.M. will be

a. exactly 4.

b. at most 2.

c. between 4 and 10, inclusive.