1. M16 Poisson Distribution 1  Department of ISM, University of Alabama, 1995-2003 Lesson Objectives  Learn when to use the Poisson distribution.  Learn how to calculate probabilities for the Poisson using the formula and the two tables in the book.  Understand the inverse relationship between the Poisson and exponential.

2. M16 Poisson Distribution 2  Department of ISM, University of Alabama, 1995-2003 A data distribution used to model the count of the number of occurrences of some event over a specified span of time, space, or distance. Poisson Distribution: Quantitative, discrete

3. Examples of Poisson Variables  Number of tornados striking Alabama per week during next spring (March, April, May).  Number of flaws in the next 100 sq. yd. of fabric produced at a textile mill  Number of potholes per mile on city streets.  Number of incoming calls to a 911 switchboard during a one-day period  Number of customers arriving at a store in a one-hour period.

4. M16 Poisson Distribution 4  Department of ISM, University of Alabama, 1995-2003 Poisson Distribution Fixed span of time, space, or distance. X = count of number of occurrences of event Possible values of a Poisson variable: 0, 1, 2, 3, …,  (whole numbers) One parameter: the average number of occurrences in the specified span of time, space, or distance Notation: X ~ Poisson(  = Mean )

5. M16 Poisson Distribution 5  Department of ISM, University of Alabama, 1995-2003 Additional Poisson Assumptions:  The number of occurrences in one interval is independent of the number in any other non-overlapping interval.  The average number of occurrences in an interval is proportional to the size of the interval.  Two or more events can’t occur at the same time or place.

6. M16 Poisson Distribution 6  Department of ISM, University of Alabama, 1995-2003 Are these Poisson variables? Number of children in a family? Number of cars passing through an intersection in 5 minutes? Number of hits on your Web site in 24 hours? Number of birdies in a round of golf?

7. M16 Poisson Distribution 7  Department of ISM, University of Alabama, 1995-2003 The probability that an event will occur exactly x times in a given span of time, space, or distance is: The Poisson distribution:

8. M16 Poisson Distribution 8  Department of ISM, University of Alabama, 1995-2003  Poi = is population std. dev. If a population of X values follows a Poisson( ) distribution, then:  Poi = is population mean Parameter for the Poisson See formula sheet

9. M16 Poisson Distribution 9  Department of ISM, University of Alabama, 1995-2003 Example: Phone calls arrive at a switchboard at an average rate of 2.0 calls per minute. If the number of calls in any time interval follows the Poisson distribution, then X = number of phone calls in a given minute. X ~ Poisson ( = Y = number of phone calls in a given hour. Y ~ Poisson ( = W = number of phone calls in a 15 seconds. W ~ Poisson ( =

10. M16 Poisson Distribution 10  Department of ISM, University of Alabama, 1995-2003 a. Find the probability of exactly five calls in the next three minutes? Y = number of calls in next three minutes 7776.002479 120 Y ~ Poisson ( = P(Y = 5) = = = See page 902, Table A.3,  = 6.0, k = 5.

11. b. Find the probability of at least two phone calls in the next three minutes? P(x  2) = 1.0 – [ P(x = 0) + P(x = 1)] 6 0 e –6 0! 1.002479 1 x: 0 1 2 3 4 5 6  6 1 e –6 1! =.982647 6.002479 1 = 6.0 want don’t want P( x = 0) = = =.002479 P( x = 1) = = =.014874 P(x  2) = 1.0 – [.002479 +.014874) ] p. 902, Table A.3,  = 6.0, k = 0, 1. p. 902, Table A.4,  = 6.0, k = 1, for P(x  1)

12. M16 Poisson Distribution 12  Department of ISM, University of Alabama, 1995-2003 Table A3, page 901-904 ; Individual Table A4, page 901-904 ; Cumulative  = 4.1 P(X = 3) = _________  = 4.1 P(X  3) = = _________ P(X=0) + P(X=1) + P(X=2) + P(X=3) Poisson Tables

13. M16 Poisson Distribution 13  Department of ISM, University of Alabama, 1995-2003 Simulation: The relationship between the Poisson and Exponential distributions. Situation: The time at which a web site receives a “hit” is randomly generated over a 96 minute period. Y = the “time between hits.” Y ~ Exponential(  = 4 min./ hit)

14. M16 Poisson Distribution 14  Department of ISM, University of Alabama, 1995-2003 1.377 1.377 6.558 7.934 0.692 8.626 0.954 9.580 1.33610.916 2.08212.998 5.58918.587 11.02329.610 18.63248.243 0.23548.478 0.09848.576 0.17248.748 4.10052.848 4.99557.843 3.99561.837 1.98263.819 7.27471.093...... X = 25 hits Y = 3.84 min./hit s Y = 4.34 min./hit X = 25 hits/ 96 min. X =.260 hits/ 1 min. Y = Time Intervals Y ~ Exp(  = 4.0 min/hit ) X = Count of “Hits” X ~ Poi( =.25 hits/min.) Elapsed Time Time Intervals 1 1 3 1 1 0 0 1 0 0 0 0 4 1 1 2 0 1 1 1 0 3 0 3 W = 1.042 hits/ 4 min. Sample Exponential Distribution Sample Poisson Distribution

15. M16 Poisson Distribution 15  Department of ISM, University of Alabama, 1995-2003 1.625 5.811 7.436 4.25811.694 1.24812.943 1.86914.812 6.32721.139 0.10721.246 2.20223.448 5.72629.174 2.43931.613 1.96133.574 4.17237.746 1.96639.712 0.96840.680 0.96041.640 0.31541.956 2.08644.041...... X = 29 hits Y = 3.12 min./hit s Y = 2.56 min./hit X = 29 hits/ 96 min. X =.302 hits/ 1 min. X = Count of Occ. X ~ Poi( =.25 hits/min.) Elapsed Time Time Intervals W = 1.208 hits/ 4 min. Sample Exponential Distribution 1 1 1 2 0 3 0 2 1 2 3 1 1 1 1 0 1 3 1 1 1 0 2 0 Sample Poisson Distribution Y = Time Intervals Y ~ Exp(  = 4.0 min/hit )

16. M16 Poisson Distribution 16  Department of ISM, University of Alabama, 1995-2003 1.625 0.138 1.763 17.34019.099 1.98621.085 0.69221.778 2.14423.921 3.44127.363 3.90631.268 1.18832.457 4.73237.189 1.89239.081 1.24940.330 3.40943.739 12.77756.516 3.59660.112 0.11960.231 2.71862.949...... X = 25 hits Y = 3.79 min./hit s Y = 4.46 min./hit X = 25 hits/ 96 min. X =.260 hits/ 1 min. Y = Time Intervals Y ~ Exp(  = 4.0 min/hit ) X = Count of Occ. X ~ Poi( =.25 hits/min.) Elapsed Time Time Intervals W = 1.042 hits/ 4 min. Sample Exponential Distribution 2 0 0 0 1 3 1 1 1 2 2 0 0 0 1 4 0 1 3 0 0 2 0 1 Sample Poisson Distribution

17. M16 Poisson Distribution 17  Department of ISM, University of Alabama, 1995-2003  In symbols: X = number of arrivals ~ Poisson( ) Y = time between arrivals ~ exponential( 1/ )  If the number of “occurrences” in a fixed interval has a Poisson distribution, then the times between “occurrences” have an exponential distribution. The mean of the exponential is the inverse of the mean of the Poisson Relationship between Poisson and Exponential

18. M16 Poisson Distribution 18  Department of ISM, University of Alabama, 1995-2003 The End