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Unformatted text preview: Chapter 5 Discrete Probability Distributions Learning Objectives 1. Understand the concepts of a random variable and a probability distribution. 2. Be able to distinguish between discrete and continuous random variables. 3. Be able to compute and interpret the expected value, variance, and standard deviation for a discrete random variable. 4. Be able to compute and work with probabilities involving a binomial probability distribution. 5. Be able to compute and work with probabilities involving a Poisson probability distribution. 6. Know when and how to use the hypergeometric probability distribution. Solutions: 5  1 Chapter 5 1. a. Head, Head (H,H) Head, Tail (H,T) Tail, Head (T,H) Tail, Tail (T,T) b. x = number of heads on two coin tosses c. Outcome Values of x (H,H) 2 (H,T) 1 (T,H) 1 (T,T) d. Discrete. It may assume 3 values: 0, 1, and 2. 2. a. Let x = time (in minutes) to assemble the product. b. It may assume any positive value: x > 0. c. Continuous 3. Let Y = position is offered N = position is not offered a. S = {(Y,Y,Y), (Y,Y,N), (Y,N,Y), (Y,N,N), (N,Y,Y), (N,Y,N), (N,N,Y), (N,N,N)} b. Let N = number of offers made; N is a discrete random variable. c. Experimental Outcome (Y,Y,Y) (Y,Y,N) (Y,N,Y) (Y,N,N) (N,Y,Y) (N,Y,N) (N,N,Y) (N,N,N) Value of N 3 2 2 1 2 1 1 4. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 5. a. S = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3)} b. Experimental Outcome (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) Number of Steps Required 2 3 4 3 4 5 6. a. values: 0,1,2,...,20 discrete b. values: 0,1,2,... discrete c. values: 0,1,2,...,50 discrete d. values: 0 ≤ x ≤ 8 continuous e. values: x > 5  2 Discrete Probability Distributions continuous 7. a. f ( x ) ≥ 0 for all values of x . Σ f ( x ) = 1 Therefore, it is a proper probability distribution. b. Probability x = 30 is f (30) = .25 c. Probability x ≤ 25 is f (20) + f (25) = .20 + .15 = .35 d. Probability x > 30 is f (35) = .40 8. a. x f ( x ) 1 3/20 = .15 2 5/20 = .25 3 8/20 = .40 4 4/20 = .20 Total 1.00 b. c. f ( x ) ≥ 0 for x = 1,2,3,4. Σ f ( x ) = 1 9. a. Age Number of Children f ( x ) 6 37,369 0.018 7 87,436 0.043 8 160,840 0.080 9 239,719 0.119 5  3 .1 .2 .3 .4 f ( x ) x 1 2 3 4 Chapter 5 10 286,719 0.142 11 306,533 0.152 12 310,787 0.154 13 302,604 0.150 14 289,168 0.143 2,021,175 1.001 b. c. f ( x ) ≥ 0 for every x Σ f ( x ) = 1 Note: Σ f ( x ) = 1.001 in part (a); difference from 1 is due to rounding values of f ( x ). 10. a. x f ( x ) 1 0.05 2 0.09 3 0.03 4 0.42 5 0.41 1.00 b. x f ( x ) 1 0.04 2 0.10 3 0.12 4 0.46 5 0.28 1.00 5  4 6 7 8 9 10 11 12 13 14 .02 .04 .06 .08 .10 .12 .14 .16 f ( x ) x Discrete Probability Distributions c. P (4 or 5) = f (4) + f (5) = 0.42 + 0.41 = 0.83 d. Probability of very satisfied: 0.28 e. Senior executives appear to be more satisfied than middle managers. 83% of senior executives have a score of 4 or 5 with 41% reporting a 5. Only 28% of middle managers report being very satisfied....
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 Spring '11
 Burns
 Probability theory, Binomial distribution, Discrete probability distribution, σ, Discrete Probability Distributions

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