1 The Poisson PD

2 Are you familiar with the rock group Aerosmith? They have a song called Toys in the Attic. If they had a song called Toys on the Roof, then the “Toys on” part rhymes with the name Poisson when you say “Toys on” real fast. Can you say Poisson? OKAY, I know this is a silly way to start this section, but I wanted to have you be able to say Poisson correctly and “Toys on” said real fast is all I could come up with. The Poisson PD is useful when certain types of problems are encountered. Some examples might be the number of people who arrive at a Dairy Queen each hour, or the number of phone calls that come into a bank card center each fifteen minute interval. The number of occurrences is a random variable that can be described by the Poisson PD.

3 If, in an experiment, 1. the probability of an occurrence is the same for any two intervals of equal length, and 2. the occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval, then the number of occurrences can be described by the Poisson PD. If x is the number of occurrences in the interval, then P(X) is the probability of x occurrences. Now P(X) = e -λ λ X /X!, where λ = the expected value or mean number of occurrences, and e =

4 Example: Say we are interested in the number of arrivals at a drive- up teller window during a 1-minute period during the noon hour. Say from past experience we know that on average 3 cars drive up to the window in a 1 minute period. Table E.7 has information about various mean amounts during a interval. Since our λ = 3 we look in that column: XP(X) And so on. On the next slide I have produced the P(X) column in Excel and I have put in the cumulative probability column.

5 Mean Number of Occurrences3 Number of ArrivalsP(X) Cumulative Probability E

6 What does the Cum Prob on the previous screen do for us? It helps us answer questions like what is the probability that 5 or fewer arrivals occur in the interval? We might write this P(X≤5)? So, P(x≤5) = What about P(x>5)? This equals 1 - P(x≤5) =.0839 Note on the previous screen that the Cum Prob does not equal 1 when x = 12. The reason is that I just listed up to 12 arrivals, but I could have listed more. In problems in this section they do something peculiar. They might say something like on average 48 calls arrive per hour. How many calls on average come in 5 minutes? Since 5 minutes is 1/12 of an hour, divide the 48 by 12, or 4. IN OTHER WORDS, pay attention to the time interval and average occurrences per time interval.

7 In excel the Poisson function has the form =poisson(number of arrivals, mean number of arrivals, false or true) The number of arrivals is in the x column, the mean number of arrivals is at the top of the table, and use false if you want P(X) or use true if you want the cum. prob. You could just use the table in the book.

Problem 22 page From Table E.7 column 6.0 XP(X) a. P(X < 5) = P(0) + P(1) + P(2) + P(3) P(4) = b. P(5) = c. P(X≥5) = 1 – P(X<5) = d. P(4 or 5) = P(4) + P(5) = question 37 page 201 And so on. Discard if 3 or less – P(X≤3) = P(0) + P(1) + P(2) + P(3) =.1512 They can expect to discard.1512(100) = cookies