LESSON 11 THE POISSON DISTRIBUTION Outline The context

LESSON 11: THE POISSON DISTRIBUTION Outline • • The context The properties Notation Formula Use of table Use of Excel Mean and variance 1

POISSON PROCESS THE CONTEXT • An important property of the Poisson process: – The probability of two or more successes will occur in an interval approaches zero as the interval becomes smaller. • Examples where interval is a length of time – Number of people coming to a bank per hour – Number of calls received by a telephone operator per hour – Number of accidents per week • Examples where interval is a length of an item – Number of defects per hundred meter 2

POISSON PROCESS THE PROPERTIES • The Poisson process has the following properties: 1. The number of successes of various intervals are independent. A Poisson process has no memory. 2. The probability that a success will occur in an interval is the same for all intervals of equal size and is proportional to the size of the interval. The mean process rate must remain constant for the entire time span pr space considered. 3. The probability of two or more successes will occur in an interval approaches zero as the interval becomes smaller. 3

POISSON DISTRIBUTION THE NOTATION • Notation – : the mean rate of success period – t : the time span (or space) considered – x : the number of observed successes • A constant The Excel function for ex is EXP(x) 4

POISSON DISTRIBUTION THE PROBABILITY DISTRIBUTION • The Poisson probability distribution gives the probability of getting exactly x successes during a given time interval or in a specified region • The probability of getting exactly x successes during a given time interval or in a specified region is as follows: 5

POISSON DISTRIBUTION THE PROBABILITY DISTRIBUTION Example 1: The marketing manager of a company has noted that she usually receives 15 complaint calls from customers during a week (consisting of 5 working days) and that the calls occur at random. Find the probability of her receiving exactly 5 calls in a single day. 6

POISSON DISTRIBUTION THE PROBABILITY DISTRIBUTION Example 2: The marketing manager of a company has noted that she usually receives 15 complaint calls from customers during a week (consisting of 5 working days) and that the calls occur at random. Find the probability of her receiving at most 2 calls in a single day. 7

POISSON DISTRIBUTION USE OF TABLE • Table C, Appendix A, pp. 533 -535 gives the probability of getting at most x successes given that t is the average number of successes occurring in the given time interval or region • The table can be used to find the probability of – exactly x successes: – at least x successes: – successes between a and b: 8

POISSON DISTRIBUTION USE OF TABLE Example 3: Find the following using Table C: Example 4: Find the following using Table C: 9

POISSON DISTRIBUTION USE OF EXCEL • The Excel function POISSON gives and • It takes three arguments, the first 2 being x and t • The last one is TRUE for and FALSE for Example 5 (self study): Find and Verify if Excel gives the same answer as it was obtained before. Answer: = POISSON(2, 3, TRUE) = POISSON(2, 3, FALSE) 10

POISSON DISTRIBUTION MEAN AND VARIANCE • If X is a Poisson random variable, the mean and the variance of X are: • • E(X) is the mean or expected value of X V(X) is the variance of X is the mean rate of successes period t is the length of time (or space) considerd 11

READING AND EXERCISES Lesson 11 Reading: Section 8 -1, pp. 228 -233 Exercises: 8 -1, 8 -2 12