1. The Poisson Distribution

2. The Poisson Distribution may be used as an approximation for a binomial distribution when n is large and p is small enough that np is of moderate size. The Poisson Distribution, with some parameter m, looks like:

3. Some examples of random variables that usually obey the Poisson Distribution: 1. The number of misprints on a page of a book. 2. The number of people in a community living to 100 years of age. 3. The number of wrong telephone numbers that are dialed in a day. 4. The number of customers entering a post office on a given day.

4. Mean and Variance of a Poisson Distribution Mean: E(X) = m Variance: Var(X) = m

5. Given a Poisson distribution and that m = 2, calculate: Answer: 0.271 Answer: 0.677 Answer: 0.594

6. Suppose that the number of typographical errors on a single page of a book has a Poisson distribution with a parameter of. Calculate the probability that there is at least one error on a page. Answer: 0.393 Suppose that the probability that an item produced by a certain machine will be defective is 0.1. Find the probability that a sample of 10 items will contain at most 1 defective item. Solve this using Binomial distribution Solve this using Poisson distribution Answer: 0.7361 Answer: 0.7358

7. A research project found that an important role in the bonding process of river otters is social grooming. After extensive observations, it is found that one group of river otters under study had a frequency of grooming of approximately 1.7 for each 10 minutes. Suppose you are observing river otters for 30 minutes. Let r = 0, 1, 2, … be a random variable that represents the number of times in a 30 minute interval that one otter grooms another. Find the probability that one otter will groom another four or more times during the 30-minute observation period. Answer: 0.7487

8. Police department officials indicate that the average number of homicides per day in New York City is 5.4. Assuming a Poisson process, what is the probability that on any given day there will be at most four homicides? Answer: 0.3733 The director of the Social Services agency for one city receives an average of five complaints a day about lack of child support claims received. Assuming it follows a Poisson distribution, what is the probability that on any day the director will receive at least three lack of child support claims? Answer: 0.8754

9. The random variable X has a Poisson distribution with mean 4. Calculate (a) (b) (c) Answers: (a) 0.547 (b) 0.762 (c) 0.718 SPEC/HL1/17

10. The number of fish present in a given stretch of river has a Poisson distribution with mean 3.8. Find the probability that in this stretch there are at most three fish. Answer: 0.473 The number of incorrectly dialed numbers made per hour by a telephone sales representative has a Poisson distribution. The probability that she dials all of the numbers correctly is 0.82. Find the mean number of incorrectly dialed numbers per hour. Answer: 0.198

11. The number of car accidents occurring per day on a highway follows a Poisson distribution with mean 1.5. a) Find the probability that more than two accidents will occur on a given Monday. b) Given that at least one accident occurs on another day, find the probability that more than two accidents occur on that day. Answers: a) 0.191 b) 0.246

12. The random variable X follows a Poisson distribution. Given that, find: a) the mean of the distribution; b) Answers: a) 2.99 b) 0.424