1. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-1 Business Statistics, 3e by Ken Black Chapter 5 Discrete Distributions

2. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-2 Learning Objectives Distinguish between discrete random variables and continuous random variables. Know how to determine the mean and variance of a discrete distribution. Identify the type of statistical experiments that can be described by the binomial distribution, and know how to work such problems.

3. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-3 Learning Objectives -- Continued Decide when to use the Poisson distribution in analyzing statistical experiments, and know how to work such problems. Decide when binomial distribution problems can be approximated by the Poisson distribution, and know how to work such problems. Decide when to use the hypergeometric distribution, and know how to work such problems.

4. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-4 Discrete vs Continuous Distributions Random Variable -- a variable which contains the outcomes of a chance experiment Discrete Random Variable -- the set of all possible values is at most a finite or a countably infinite number of possible values –Number of new subscribers to a magazine –Number of bad checks received by a restaurant –Number of absent employees on a given day Continuous Random Variable -- takes on values at every point over a given interval –Current Ratio of a motorcycle distributorship –Elapsed time between arrivals of bank customers –Percent of the labor force that is unemployed

5. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-5 Some Special Distributions Discrete –binomial –Poisson –hypergeometric Continuous –normal –uniform –exponential –t –chi-square –F

6. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-6 Discrete Distribution -- Example 012345012345 0.37 0.31 0.18 0.09 0.04 0.01 Number of Crises Probability Distribution of Daily Crises 0 0.1 0.2 0.3 0.4 0.5 012345 ProbabilityProbability Number of Crises

7. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-7 Requirements for a Discrete Probability Function Probabilities are between 0 and 1, inclusively Total of all probabilities equals 1

8. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-8 Requirements for a Discrete Probability Function -- Examples XP(X) 0 1 2 3.1.2.4.2.1 1.0 XP(X) 0 1 2 3 -.1.3.4.3.1 1.0 XP(X) 0 1 2 3.1.3.4.3.1 1.2

9. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-9 Mean of a Discrete Distribution X 0 1 2 3 P(X).1.2.4.2.1 -.1.0.4.3 1.0 XPX  ()

10. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-10 Variance and Standard Deviation of a Discrete Distribution X 0 1 2 3 P(X).1.2.4.2.1 -2 0 1 2 X  4101441014.4.2.0.2.4 1.2

11. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-11 Mean of the Crises Data Example XP(X) X  0.37.00 1.31 2.18.36 3.09.27 4.04.16 5.01.05 1.15 0 0.1 0.2 0.3 0.4 0.5 012345 ProbabilityProbability Number of Crises

12. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-12 Variance and Standard Deviation of Crises Data Example XP(X) (X-  )  ) 2  ) 2  P(X) 0.37-1.151.32.49 1.31-0.150.02.01 2.180.850.72.13 3.091.853.42.31 4.042.858.12.32 5.013.8514.82.15 1.41

13. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-13 Binomial Distribution Experiment involves n identical trials Each trial has exactly two possible outcomes: success and failure Each trial is independent of the previous trials –p is the probability of a success on any one trial –q = (1-p) is the probability of a failure on any one trial –p and q are constant throughout the experiment –X is the number of successes in the n trials Applications –Sampling with replacement –Sampling without replacement -- n < 5% N

14. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-14 Binomial Distribution Probability function Mean value Variance and standard deviation

15. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-15 Binomial Distribution: Development Experiment: randomly select, with replacement, two families from the residents of Tiny Town Success is ‘Children in Household:’ p = 0.75 Failure is ‘No Children in Household:’ q = 1- p = 0.25 X is the number of families in the sample with ‘Children in Household’ Family Children in Household Number of Automobiles ABCDABCD Yes No Yes 32123212 Listing of Sample Space (A,B), (A,C), (A,D), (D,D), (B,A), (B,B), (B,C), (B,D), (C,A), (C,B), (C,C), (C,D), (D,A), (D,B), (D,C), (D,D)

16. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-16 Binomial Distribution: Development Continued Families A, B, and D have children in the household; family C does not Success is ‘Children in Household:’ p = 0.75 Failure is ‘No Children in Household:’ q = 1- p = 0.25 X is the number of families in the sample with ‘Children in Household’ (A,B), (A,C), (A,D), (D,D), (B,A), (B,B), (B,C), (B,D), (C,A), (C,B), (C,C), (C,D), (D,A), (D,B), (D,C), (D,D) Listing of Sample Space 21222212110122122122221211012212 X 1/16 P(outcome)

17. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-17 Binomial Distribution: Development Continued (A,B), (A,C), (A,D), (D,D), (B,A), (B,B), (B,C), (B,D), (C,A), (C,B), (C,C), (C,D), (D,A), (D,B), (D,C), (D,D) Listing of Sample Space 21222212110122122122221211012212 X 1/16 P(outcome) X 012012 1/16 6/16 9/16 1 P(X)

18. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-18 Binomial Distribution: Development Continued Families A, B, and D have children in the household; family C does not Success is ‘Children in Household:’ p = 0.75 Failure is ‘No Children in Household:’ q = 1- p = 0.25 X is the number of families in the sample with ‘Children in Household’ X Possible Sequences 01120112 (F,F) (S,F) (F,S) (S,S) P(sequence) (.) ) (.) 25 2  (.) )2575 (.) )7525 (.) ) (.) 75 2 

19. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-19 Binomial Distribution: Development Continued X Possible Sequences 01120112 (F,F) (S,F) (F,S) (S,S) P(sequence) (.) ) (.) 25 2  (.) )2575 (.) )7525 (.) ) (.) 75 2  X 012012 P(X) (.) )2575 2 =0.375 (.) ) (.) 75 2  =0.5625 (.) ) (.) 25 2  =0.0625

20. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-20 Binomial Distribution: Demonstration Problem 5.3

21. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-21 Binomial Table n = 20PROBABILITY X0.10.20.30.40.50.60.70.80.9 00.1220.0120.0010.000 10.2700.0580.0070.000 20.2850.1370.0280.0030.000 30.1900.2050.0720.0120.0010.000 40.0900.2180.1300.0350.0050.000 50.0320.1750.1790.0750.0150.0010.000 60.0090.1090.1920.1240.0370.0050.000 70.0020.0550.1640.1660.0740.0150.0010.000 8 0.0220.1140.1800.1200.0350.0040.000 9 0.0070.0650.160 0.0710.0120.000 100.0000.0020.0310.1170.1760.1170.0310.0020.000 110.000 0.0120.0710.160 0.0650.0070.000 120.000 0.0040.0350.1200.1800.1140.0220.000 130.000 0.0010.0150.0740.1660.1640.0550.002 140.000 0.0050.0370.1240.1920.1090.009 150.000 0.0010.0150.0750.1790.1750.032 160.000 0.0050.0350.1300.2180.090 170.000 0.0010.0120.0720.2050.190 180.000 0.0030.0280.1370.285 190.000 0.0070.0580.270 200.000 0.0010.0120.122

22. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-22 Using the Binomial Table Demonstration Problem 5.4 n = 20PROBABILITY X0.10.20.30.4 00.1220.0120.0010.000 10.2700.0580.0070.000 20.2850.1370.0280.003 30.1900.2050.0720.012 40.0900.2180.1300.035 50.0320.1750.1790.075 60.0090.1090.1920.124 70.0020.0550.1640.166 80.0000.0220.1140.180 90.0000.0070.0650.160 100.0000.0020.0310.117 110.000 0.0120.071 120.000 0.0040.035 130.000 0.0010.015 140.000 0.005 150.000 0.001 160.000 170.000 180.000 190.000 200.000

23. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-23 Binomial Distribution using Table: Demonstration Problem 5.3 n = 20PROBABILITY X0.050.060.07 00.35850.29010.2342 10.37740.37030.3526 20.18870.22460.2521 30.05960.08600.1139 40.01330.02330.0364 50.00220.00480.0088 60.00030.00080.0017 70.00000.00010.0002 80.0000 ………… 200.0000

24. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-24 Graphs of Selected Binomial Distributions n = 4PROBABILITY X0.10.50.9 00.6560.0630.000 10.2920.2500.004 20.0490.3750.049 30.0040.2500.292 40.0000.0630.656 P = 0.1 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 01234 X P(X) P = 0.5 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 01234 X P(X) P = 0.9 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 01234 X P(X)

25. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-25 Poisson Distribution Describes discrete occurrences over a continuum or interval A discrete distribution Describes rare events Each occurrence is independent any other occurrences. The number of occurrences in each interval can vary from zero to infinity. The expected number of occurrences must hold constant throughout the experiment.

26. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-26 Poisson Distribution: Applications Arrivals at queueing systems –airports -- people, airplanes, automobiles, baggage –banks -- people, automobiles, loan applications –computer file servers -- read and write operations Defects in manufactured goods –number of defects per 1,000 feet of extruded copper wire –number of blemishes per square foot of painted surface –number of errors per typed page

27. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-27 Poisson Distribution Probability function nMean value nStandard deviation nVariance

28. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-28 Poisson Distribution: Demonstration Problem 5.7

29. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-29 Poisson Distribution: Probability Table X0.51.51.63.03.26.46.57.08.0 00.60650.22310.20190.04980.04080.00170.00150.00090.0003 10.30330.33470.32300.14940.13040.01060.00980.00640.0027 20.07580.25100.25840.22400.20870.03400.03180.02230.0107 30.01260.12550.13780.22400.22260.07260.06880.05210.0286 40.00160.04710.05510.16800.17810.11620.11180.09120.0573 50.00020.01410.01760.10080.11400.14870.14540.12770.0916 60.00000.00350.00470.05040.06080.15860.15750.14900.1221 70.00000.00080.00110.02160.02780.14500.14620.14900.1396 80.00000.00010.00020.00810.01110.11600.11880.13040.1396 90.0000 0.00270.00400.08250.08580.10140.1241 100.0000 0.00080.00130.05280.05580.07100.0993 110.0000 0.00020.00040.03070.03300.04520.0722 120.0000 0.0001 0.01640.01790.02630.0481 130.0000 0.00810.00890.01420.0296 140.0000 0.00370.00410.00710.0169 150.0000 0.00160.00180.00330.0090 160.0000 0.00060.00070.00140.0045 170.0000 0.00020.00030.00060.0021 180.0000 0.0001 0.00020.0009

30. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-30 Poisson Distribution: Using the Poisson Tables X0.51.51.63.0 00.60650.22310.20190.0498 10.30330.33470.32300.1494 20.07580.25100.25840.2240 30.01260.12550.13780.2240 40.00160.04710.05510.1680 50.00020.01410.01760.1008 60.00000.00350.00470.0504 70.00000.00080.00110.0216 80.00000.00010.00020.0081 90.0000 0.0027 100.0000 0.0008 110.0000 0.0002 120.0000 0.0001

31. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-31 Poisson Distribution: Using the Poisson Tables X0.51.51.63.0 00.60650.22310.20190.0498 10.30330.33470.32300.1494 20.07580.25100.25840.2240 30.01260.12550.13780.2240 40.00160.04710.05510.1680 50.00020.01410.01760.1008 60.00000.00350.00470.0504 70.00000.00080.00110.0216 80.00000.00010.00020.0081 90.0000 0.0027 100.0000 0.0008 110.0000 0.0002 120.0000 0.0001

32. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-32 Poisson Distribution: Using the Poisson Tables X0.51.51.63.0 00.60650.22310.20190.0498 10.30330.33470.32300.1494 20.07580.25100.25840.2240 30.01260.12550.13780.2240 40.00160.04710.05510.1680 50.00020.01410.01760.1008 60.00000.00350.00470.0504 70.00000.00080.00110.0216 80.00000.00010.00020.0081 90.0000 0.0027 100.0000 0.0008 110.0000 0.0002 120.0000 0.0001

33. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-33 Poisson Distribution: Graphs 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 012345678 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0246810121416

34. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-34 Poisson Approximation of the Binomial Distribution Binomial probabilities are difficult to calculate when n is large. Under certain conditions binomial probabilities may be approximated by Poisson probabilities. Poisson approximation

35. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-35 Poisson Approximation of the Binomial Distribution XError 00.22310.2181-0.0051 10.33470.33720.0025 20.25100.25550.0045 30.12550.12640.0009 40.04710.0459-0.0011 50.01410.0131-0.0010 60.00350.0030-0.0005 70.00080.0006-0.0002 80.0001 0.0000 9 XError 00.0498 0.0000 10.14940.14930.0000 20.22400.22410.0000 30.22400.22410.0000 40.16800.16810.0000 50.1008 0.0000 60.0504 0.0000 70.0216 0.0000 80.0081 0.0000 90.0027 0.0000 100.0008 0.0000 110.0002 0.0000 120.0001 0.0000 130.0000

36. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-36 Hypergeometric Distribution Sampling without replacement from a finite population The number of objects in the population is denoted N. Each trial has exactly two possible outcomes, success and failure. Trials are not independent X is the number of successes in the n trials The binomial is an acceptable approximation, if n < 5% N. Otherwise it is not.

37. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-37 Hypergeometric Distribution Probability function –N is population size –n is sample size –A is number of successes in population –x is number of successes in sample Mean value Variance and standard deviation

38. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-38 Hypergeometric Distribution: Probability Computations N = 24 X = 8 n = 5 x 00.1028 10.3426 20.3689 30.1581 40.0264 50.0013 P(x)

39. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-39 Hypergeometric Distribution: Graph N = 24 X = 8 n = 5 x 00.1028 10.3426 20.3689 30.1581 40.0264 50.0013 P(x) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 012345

40. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-40 Hypergeometric Distributio: Demonstration Problem 5.11 XP(X) 00.0245 10.2206 20.4853 3 0.2696 N = 18 n = 3 A = 12

41. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-41 Hypergeometric Distribution: Binomial Approximation (large n) Hypergeometric N = 24 X = 8 n = 5 Binomial n = 5 p = 8/24 =1/3 xError 00.10280.1317 -0.0289 10.34260.3292 0.0133 20.36890.3292 0.0397 30.15810.1646 -0.0065 40.02640.0412 -0.0148 50.00130.0041 -0.0028 P(x)

42. Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 5-42 Hypergeometric Distribution: Binomial Approximation (small n) Hypergeometric N = 240 X = 80 n = 5 Binomial n = 5 p = 80/240 =1/3 xP(x)Error 00.12890.1317 -0.0028 10.33060.3292 0.0014 20.33270.3292 0.0035 30.16420.1646 -0.0004 40.03980.0412 -0.0014 50.00380.0041 -0.0003 P(x)