1. 4 - 1 © 1998 Prentice-Hall, Inc. Statistics for Business & Economics Discrete Random Variables Chapter 4

2. 4 - 2 © 1998 Prentice-Hall, Inc. Learning Objectives 1.Define random variable 2.Compute the expected value & variance of discrete random variables 3.Describe the binomial & Poisson probability distributions 4.Calculate probabilities for binomial & Poisson random variables

3. 4 - 3 © 1998 Prentice-Hall, Inc. Thinking Challenge You’re taking a 33 question multiple choice test. Each question has 4 choices. Clueless on 1 question, you decide to guess. What’s the chance you’ll get it right? If you guessed on all 33 questions, what would be your grade? Pass? AloneGroupClass

4. 4 - 4 © 1998 Prentice-Hall, Inc. Random Variable 1.A numerical outcome of an experiment 2.May be discrete or continuous 3.Discrete random variable Countable number of values Countable number of values Example: Number of tails in 2 coin tosses Example: Number of tails in 2 coin tosses 4.Continuous random variable Infinite number of values within an interval Infinite number of values within an interval Example: Amount of soda in a 12 oz. can Example: Amount of soda in a 12 oz. can

5. 4 - 5 © 1998 Prentice-Hall, Inc. Discrete Random Variables

6. 4 - 6 © 1998 Prentice-Hall, Inc. Discrete Random Variable 1. Type of random variable 2.Whole number (0, 1, 2, 3 etc.) 3.Obtained by counting 4.Usually finite number of values Poisson random variable is exception (  ) Poisson random variable is exception (  )

7. 4 - 7 © 1998 Prentice-Hall, Inc. Discrete Random Variable Examples

8. 4 - 8 © 1998 Prentice-Hall, Inc. Discrete Random Variable Examples Random Variable Possible Values Experiment

9. 4 - 9 © 1998 Prentice-Hall, Inc. Discrete Random Variable Examples Random Variable Possible Values Make 100 sales calls Experiment

10. 4 - 10 © 1998 Prentice-Hall, Inc. Discrete Random Variable Examples Random Variable Possible Values Make 100 sales calls # Sales 0, 1, 2,..., 100 Experiment

11. 4 - 11 © 1998 Prentice-Hall, Inc. Discrete Random Variable Examples Random Variable Possible Values Make 100 sales calls # Sales 0, 1, 2,..., 100 Inspect 70 radios Experiment

12. 4 - 12 © 1998 Prentice-Hall, Inc. Discrete Random Variable Examples Random Variable Possible Values Make 100 sales calls # Sales 0, 1, 2,..., 100 Inspect 70 radios # Defective 0, 1, 2,..., 70 Experiment

13. 4 - 13 © 1998 Prentice-Hall, Inc. Discrete Random Variable Examples Random Variable Possible Values Make 100 sales calls # Sales 0, 1, 2,..., 100 Inspect 70 radios # Defective 0, 1, 2,..., 70 Answer 33 questions Experiment

14. 4 - 14 © 1998 Prentice-Hall, Inc. Discrete Random Variable Examples Random Variable Possible Values Make 100 sales calls # Sales 0, 1, 2,..., 100 Inspect 70 radios # Defective 0, 1, 2,..., 70 Answer 33 questions # Correct 0, 1, 2,..., 33 Experiment

15. 4 - 15 © 1998 Prentice-Hall, Inc. Discrete Random Variable Examples Random Variable Possible Values Make 100 sales calls # Sales 0, 1, 2,..., 100 Inspect 70 radios # Defective 0, 1, 2,..., 70 Answer 33 questions # Correct 0, 1, 2,..., 33 Count cars at toll between 11:00 & 1:00 Experiment

16. 4 - 16 © 1998 Prentice-Hall, Inc. Discrete Random Variable Examples Random Variable Possible Values Make 100 sales calls # Sales 0, 1, 2,..., 100 Inspect 70 radios # Defective 0, 1, 2,..., 70 Answer 33 questions # Correct 0, 1, 2,..., 33 Count cars at toll between 11:00 & 1:00 # Cars arriving 0, 1, 2,...,  Experiment

17. 4 - 17 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution 1.List of all possible [x, p(x)] pairs x = Value of random variable (outcome) x = Value of random variable (outcome) p(x) = Probability associated with value p(x) = Probability associated with value 2.Mutually exclusive (no overlap) 3.Collectively exhaustive (nothing left out) 4. 0  p(x)  1 (or p(x)  0) 5.  p(x) = 1

18. 4 - 18 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Example

19. 4 - 19 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Example Experiment: Toss 2 coins. Count # tails.

20. 4 - 20 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Example Probability Distribution Values, x Probabilities, p(x) Experiment: Toss 2 coins. Count # tails.

21. 4 - 21 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Example Probability Distribution Values, x Probabilities, p(x) Experiment: Toss 2 coins. Count # tails. © 1984-1994 T/Maker Co.

22. 4 - 22 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Example Probability Distribution Values, x Probabilities, p(x) 0 Experiment: Toss 2 coins. Count # tails. © 1984-1994 T/Maker Co.

23. 4 - 23 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Example Probability Distribution Values, x Probabilities, p(x) 01 Experiment: Toss 2 coins. Count # tails. © 1984-1994 T/Maker Co.

24. 4 - 24 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Example Probability Distribution Values, x Probabilities, p(x) 01 Experiment: Toss 2 coins. Count # tails. © 1984-1994 T/Maker Co.

25. 4 - 25 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Example Probability Distribution Values, x Probabilities, p(x) 012 Experiment: Toss 2 coins. Count # tails. © 1984-1994 T/Maker Co.

26. 4 - 26 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Example Probability Distribution Values, x Probabilities, p(x) 01/4 =.25 12 Experiment: Toss 2 coins. Count # tails. © 1984-1994 T/Maker Co.

27. 4 - 27 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Example Probability Distribution Values, x Probabilities, p(x) 01/4 =.25 12/4 =.50 2 Experiment: Toss 2 coins. Count # tails. © 1984-1994 T/Maker Co.

28. 4 - 28 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Example Probability Distribution Values, x Probabilities, p(x) 01/4 =.25 12/4 =.50 21/4 =.25 Experiment: Toss 2 coins. Count # tails. © 1984-1994 T/Maker Co.

29. 4 - 29 © 1998 Prentice-Hall, Inc. Visualizing Discrete Probability Distributions

30. 4 - 30 © 1998 Prentice-Hall, Inc. Visualizing Discrete Probability Distributions Listing

31. 4 - 31 © 1998 Prentice-Hall, Inc. Visualizing Discrete Probability Distributions ListingTable # Tails f(x) Count p(x) 01.25 12.50 21.25

32. 4 - 32 © 1998 Prentice-Hall, Inc. Visualizing Discrete Probability Distributions ListingTable Graph # Tails f(x) Count p(x) 01.25 12.50 21.25.00.25.50 012 x p(x)

33. 4 - 33 © 1998 Prentice-Hall, Inc. Visualizing Discrete Probability Distributions ListingTable GraphEquation # Tails f(x) Count p(x) 01.25 12.50 21.25 px n xnx pp xnx () ! !()! ()    1.00.25.50 012 x p(x)

34. 4 - 34 © 1998 Prentice-Hall, Inc. Summary Measures 1.Expected value Mean of probability distribution Mean of probability distribution Weighted average of all possible values Weighted average of all possible values  = E(X) =  x p(x)  = E(X) =  x p(x) 2.Variance Weighted average squared deviation about mean Weighted average squared deviation about mean  2 = E[ (x    (x    p(x)  2 = E[ (x    (x    p(x)

35. 4 - 35 © 1998 Prentice-Hall, Inc. Summary Measures Calculation Table xp(x)xp(x)x -  (x-  ) 2 ( x -  ) 2 p( p( x ) Total  ( x -  )2 x ) xp(x)

36. 4 - 36 © 1998 Prentice-Hall, Inc. Thinking Challenge You toss 2 coins. You’re interested in the number of tails. What are the expected value & standard deviation of this random variable, number of tails? © 1984-1994 T/Maker Co. AloneGroupClass

37. 4 - 37 © 1998 Prentice-Hall, Inc. Expected Value & Variance Solution* 0.2501.00.25 1.50.50000 2.25.501.001.00.25  = 1.0 = 1.0  2 =.50 =.50 xp(x)xp(x)x -  (x-  ) 2 ( x -  ) 2 p( p( x )

38. 4 - 38 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Function

39. 4 - 39 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Function 1.Type of model Representation of some underlying phenomenon Representation of some underlying phenomenon 2.Mathematical formula 3.Represents discrete random variable 4.Used to get exact probabilities PXx x () !  x e -

40. 4 - 40 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Models

41. 4 - 41 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Models Discrete Probability Distribution

42. 4 - 42 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Models Discrete Probability Distribution Binomial

43. 4 - 43 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Models Discrete Probability Distribution BinomialPoisson

44. 4 - 44 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Models

45. 4 - 45 © 1998 Prentice-Hall, Inc. Binomial Distribution

46. 4 - 46 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Models

47. 4 - 47 © 1998 Prentice-Hall, Inc. Binomial Random Variable 1.Number of ‘successes’ in a sample of n observations (trials)

48. 4 - 48 © 1998 Prentice-Hall, Inc. Binomial Random Variable # Reds in 15 spins of roulette wheel # Reds in 15 spins of roulette wheel # Defective items in a batch of 5 items # Defective items in a batch of 5 items # Correct on a 33 question exam # Correct on a 33 question exam # Customers who purchase out of 100 customers who enter store # Customers who purchase out of 100 customers who enter store 1.Number of ‘successes’ in a sample of n observations (trials) 2.Examples

49. 4 - 49 © 1998 Prentice-Hall, Inc. Binomial Distribution Characteristics 1.Sequence of n identical trials 2.Each trial has 2 outcomes ‘Success’ (desired outcome) or ‘failure’ ‘Success’ (desired outcome) or ‘failure’ 3.Constant trial probability 4.Trials are independent 5.Two different sampling methods Infinite population with replacement Infinite population with replacement Finite population without replacement Finite population without replacement

50. 4 - 50 © 1998 Prentice-Hall, Inc. Binomial Probability Distribution Function p(x) = Probability of x ‘successes’ in n trials n=Sample size p=Probability of ‘success’ x=Number of ‘successes’ in sample (x = 0, 1, 2,..., n)

51. 4 - 51 © 1998 Prentice-Hall, Inc. Binomial Distribution Characteristics

52. 4 - 52 © 1998 Prentice-Hall, Inc. Binomial Distribution Characteristics Mean

53. 4 - 53 © 1998 Prentice-Hall, Inc. Binomial Distribution Characteristics Mean Standard Deviation

54. 4 - 54 © 1998 Prentice-Hall, Inc. Binomial Distribution Characteristics n = 5 p = 0.1 n = 5 p = 0.5 Mean Standard Deviation

55. 4 - 55 © 1998 Prentice-Hall, Inc. Binomial Probability Distribution Example Experiment: Toss 1 coin 5 times in a row. Note # tails. What’s the probability of 3 tails?

56. 4 - 56 © 1998 Prentice-Hall, Inc. Binomial Probability Distribution Example Experiment: Toss 1 coin 5 times in a row. Note # tails. What’s the probability of 3 tails?

57. 4 - 57 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table

58. 4 - 58 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table Cumulative probabilities: p(x  k) given n & p

59. 4 - 59 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table Cumulative probabilities: p(x  k) given n & p

60. 4 - 60 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table Cumulative probabilities: p(x  k) given n & p

61. 4 - 61 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table Cumulative probabilities: p(x  k) given n & p

62. 4 - 62 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table Select table for n = 5 n = 5 p k.01…0.50….99 0.951.951….031….000 1.999.999….188….000 21.000….500….000 31.000….812….001 41.000….969….049

63. 4 - 63 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table n = 5 p k.01…0.50….99 0.951.951….031….000 1.999.999….188….000 21.000….500….000 31.000….812….001 41.000….969….049 Select row for k = 3

64. 4 - 64 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table n = 5 p k.01…0.50….99 0.951.951….031….000 1.999.999….188….000 21.000….500….000 31.000….812….001 41.000….969….049 Select column for p = 0.50

65. 4 - 65 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table n = 5 p k.01…0.50….99 0.951.951….031….000 1.999.999….188….000 21.000….500….000 31.000….812….001 41.000….969….049 Cumulative probability: p(x  3) =.812

66. 4 - 66 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table n = 5 p k.01…0.50….99 0.951.951….031….000 1.999.999….188….000 21.000….500….000 31.000….812….001 41.000….969….049 p(x  3) = p(x  3) - p(x  2). Select row for k = 2

67. 4 - 67 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table n = 5 p k.01…0.50….99 0.951.951….031….000 1.999.999….188….000 21.000….500….000 31.000….812….001 41.000….969….049 Select column for p = 0.50

68. 4 - 68 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table n = 5 p k.01…0.50….99 0.951.951….031….000 1.999.999….188….000 21.000….500….000 31.000….812….001 41.000….969….049 Cumulative probability: p(x  2) =.500

69. 4 - 69 © 1998 Prentice-Hall, Inc. Using the Binomial Probability Table n = 5 p k.01…0.50….99 0.951.951….031….000 1.999.999….188….000 21.000….500….000 31.000….812….001 41.000….969….049 p(x  3) = p(x  3) - p(x  2) =.812 -.500 =.312

70. 4 - 70 © 1998 Prentice-Hall, Inc. Binomial Distribution Thinking Challenge You’re a telemarketer selling service contracts for Macy’s. You’ve sold 20 in your last 100 calls (p =.20). If you call 12 people tonight, what’s the probability of A. No sales? B. Exactly 2 sales? C. At most 2 sales? D. At least 2 sales? AloneGroupClass

71. 4 - 71 © 1998 Prentice-Hall, Inc. Binomial Distribution Solution* Using the Binomial Formula: A. p(0) =.0687 B. p(2) =.2835 C. p(at most 2)= p(0) + p(1) + p(2) =.0687 +.2062 +.2835 =.5584 D. p(at least 2)= p(2) + p(3)...+ p(12) = 1 - [p(0) + p(1)] = 1 -.0687 -.2062 =.7251

72. 4 - 72 © 1998 Prentice-Hall, Inc. Poisson Distribution

73. 4 - 73 © 1998 Prentice-Hall, Inc. Discrete Probability Distribution Models

74. 4 - 74 © 1998 Prentice-Hall, Inc. Poisson Random Variable 1.Number of events that occur in an interval Events per unit Events per unit Time, length, area, space Time, length, area, space 2.Examples # Customers arriving in 20 minutes # Customers arriving in 20 minutes # Strikes per year in the U.S. # Strikes per year in the U.S. # Defects per lot (group) of VCR’s # Defects per lot (group) of VCR’s

75. 4 - 75 © 1998 Prentice-Hall, Inc. Poisson Process 1.Constant event probability Average of 60/hr is 1/min for 60 1-minute intervals Average of 60/hr is 1/min for 60 1-minute intervals 2.One event per interval Don’t arrive together Don’t arrive together 3.Independent events Arrival of 1 person does not affect another’s arrival Arrival of 1 person does not affect another’s arrival © 1984-1994 T/Maker Co.

76. 4 - 76 © 1998 Prentice-Hall, Inc. Poisson Probability Distribution Function p(x) = Probability of x given p(x) = Probability of x given =Expected (mean) number of ‘successes’ =Expected (mean) number of ‘successes’ e=2.71828 (base of natural logs) x=Number of ‘successes’ per unit px x () ! xe-

77. 4 - 77 © 1998 Prentice-Hall, Inc. Poisson Distribution Characteristics

78. 4 - 78 © 1998 Prentice-Hall, Inc. Poisson Distribution Characteristics Mean

79. 4 - 79 © 1998 Prentice-Hall, Inc. Poisson Distribution Characteristics Mean Standard Deviation

80. 4 - 80 © 1998 Prentice-Hall, Inc. Poisson Distribution Characteristics  = 0.5  = 6 Mean Standard Deviation

81. 4 - 81 © 1998 Prentice-Hall, Inc. Poisson Distribution Example Patients arrive at a hospital clinic at a rate of 72 per hour. What is the probability of 4 patients arriving in 3 minutes? © 1995 Corel Corp.

82. 4 - 82 © 1998 Prentice-Hall, Inc. Poisson Distribution Solution 72 per hr. = 1.2 per min. = 3.6 per 3 min. interval

83. 4 - 83 © 1998 Prentice-Hall, Inc. Poisson Distribution Solution 72 per hr. = 1.2 per min. = 3.6 per 3 min. interval

84. 4 - 84 © 1998 Prentice-Hall, Inc. Using the Poisson Probability Table

85. 4 - 85 © 1998 Prentice-Hall, Inc. Using the Poisson Probability Table Cumulative probabilities x 0…34…9.02.980… ::::::: 3.4.033….558.744….997 3.6.027….515.706….996 3.8.022….473.668….994 :::::::

86. 4 - 86 © 1998 Prentice-Hall, Inc. Using the Poisson Probability Table x 0…34…9.02.980… ::::::: 3.4.033….558.744….997 3.6.027….515.706….996 3.8.022….473.668….994 ::::::: Select row with = 3.6

87. 4 - 87 © 1998 Prentice-Hall, Inc. Using the Poisson Probability Table x 0…34…9.02.980… ::::::: 3.4.033….558.744….997 3.6.027….515.706….996 3.8.022….473.668….994 ::::::: p(x  4) = p(x  4) - p(x  3).

88. 4 - 88 © 1998 Prentice-Hall, Inc. Using the Poisson Probability Table x 0…34…9.02.980… ::::::: 3.4.033….558.744….997 3.6.027….515.706….996 3.8.022….473.668….994 ::::::: p(x  4) = p(x  4) - p(x  3). Select column x = 4.

89. 4 - 89 © 1998 Prentice-Hall, Inc. Using the Poisson Probability Table x 0…34…9.02.980… ::::::: 3.4.033….558.744….997 3.6.027….515.706….996 3.8.022….473.668….994 ::::::: p(x  4) = p(x  4) - p(x  3) =.706 - p(x  3)

90. 4 - 90 © 1998 Prentice-Hall, Inc. Using the Poisson Probability Table x 0…34…9.02.980… ::::::: 3.4.033….558.744….997 3.6.027….515.706….996 3.8.022….473.668….994 ::::::: Select column x = 3 p(x  4) = p(x  4) - p(x  3) =.706 - p(x  3)

91. 4 - 91 © 1998 Prentice-Hall, Inc. Using the Poisson Probability Table p(x  4) = p(x  4) - p(x  3) =.706 -.515 =.191 x 0…34…9.02.980… ::::::: 3.4.033….558.744….997 3.6.027….515.515.706.706….996 3.8.022….473.668….994 :::::::

92. 4 - 92 © 1998 Prentice-Hall, Inc. Thinking Challenge You work in Quality Assurance for an investment firm. A clerk enters 75 words per minute with 6 errors per hour. What is the probability of 0 errors in a 255-word bond transaction? © 1984-1994 T/Maker Co. AloneGroupClass

93. 4 - 93 © 1998 Prentice-Hall, Inc. Poisson Distribution Solution: Finding * 75 words/min = (75 words/min)(60 min/hr) = 4500 words/hr 6 errors/hr= 6 errors/4500 words 6 errors/hr= 6 errors/4500 words =.00133 errors/word In a 255-word transaction (interval): = (.00133 errors/word )(255 words) = (.00133 errors/word )(255 words) =.34 errors/255-word transaction =.34 errors/255-word transaction

94. 4 - 94 © 1998 Prentice-Hall, Inc. Poisson Distribution Solution: Finding p(0)* px x p () ! () !  x =.7118.7118e-.e-340 0f.34 0a

95. 4 - 95 © 1998 Prentice-Hall, Inc. Conclusion 1.Defined random variable 2.Computed the expected value & variance of discrete random variables 3.Described the binomial & Poisson probability distributions 4.Calculated probabilities for binomial & Poisson random variables

96. 4 - 96 © 1998 Prentice-Hall, Inc. This Class... 1.What was the most important thing you learned in class today? 2.What do you still have questions about? 3.How can today’s class be improved? Please take a moment to answer the following questions in writing:

97. End of Chapter Any blank slides that follow are blank intentionally.