2.Within a single interval, an infinite number of occurrences of the event are theoretically possible. We are not restricted to a fixed number of trials. 3.The events occur independently both within the same interval and between consecutive intervals.
91Example: Suppose X represents the number of individuals involved in a motor vehicle accident each year and X ~ Poisson with = 2.4. a.P(X = 0) = b.P(X = 1) = c.P(X = 2) = d.P(X = 3) = e.P(X = 4) = f.P(X = 5) = g.P(X > 5) =
92Center and Spread of the Poisson Distribution The mean and the variance of the Poisson distribution are identical and can be represented by the single parameter .
93Example: The number of cases of tetanus reported in the United States during a single month in 1989 has a Poisson distribution with parameter = 4.5. a.What is the probability that exactly one case of tetanus will be reported? b.What is the probability that at most two cases of tetanus will be reported? c.What is the probability that four or more cases will be reported? d.What is the mean number of cases of tetanus reported in a one-month period? What is the standard deviation?
94Poisson Approximation to the Binomial The binomial distribution with large n and small p can be accurately approximated by a Poisson distribution with parameter = np. A rule of thumb is to use the approximation when n 100 and p .01.
95Example: Let X be a random variable that represents the number of infants in a group of 500 who die before reaching their first birthdays. In the U.S., the probability that a child dies during his or her first year of life is 0.0085. a.What is the mean number of infants who would die in a group of this size? b.What is the probability that at most five infants out of 500 die in their first year of life? c.What is the probability that between 5 and 10 infants inclusive die in their first year of life?