Chap004_solutions - CHAPTER 4Discrete Random Variables 4.1...

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Unformatted text preview: CHAPTER 4Discrete Random Variables 4.1 A random variable is a variable that assumes numerical values determined by the outcome of an experiment. 4.2 The values of a discrete random variable can be counted, or listed; the values of a continuous random variable cannot be counted, or listed. 4.3 a. Discrete b. Discrete c. Continuous d. Discrete e. Discrete f. Continuous g. Continuous 4.4 See page 160 in text. 4.5 The probability of each possible outcome is > 0, and the sum of the probabilities of all possible outcomes is 1. 4.6 The mean is the sum of each possible value of x multiplied by the probability of that value of x . This represents the mean expected value of x . The standard deviation is the square root of the probability weighted sum of the squared deviations from the mean. 4.7 The standard deviation measures the spread of the population of the random variable. 4.8 a. Valid b. Not valid; p 1 2 ( 29 = 1 c. Not valid; p ( x ) = .9 all x d. Valid 4.9 a. x = 0(.2) + 1(.8) = .8 2 x = (0 .8) 2 (.2) + (1 .8) 2 (.8) = .16 x = 4 . 16 . = 43 Chapter 4: Discrete Random Variables b. x = xp ( x ) all x = 0(.25) + 1(.45) + 2(.2) + 3(.1) = 1.15 8275 . ) 1 (. ) 15 . 1 3 ( ) 2 (. ) 15 . 1 2 ( ) 45 (. ) 15 . 1 1 ( ) 25 (. ) 15 . 1 ( 2 2 2 2 2 =- +- +- +- = x 9097 . 8275 . = = x c. 6 . 1 ) 2 (. 5 ) 4 (. 2 ) 3 (. ) 1 (. 2 = + + +- = x 1071 . 2 44 . 4 44 . 4 ) 2 (. ) 6 . 1 5 ( ) 4 (. ) 6 . 1 2 ( ) 3 (. ) 6 . 1 ( ) 1 (. ) 6 . 1 2 ( 2 2 2 2 2 = = =- +- +- +-- = x x Since the probabilities sum to 1.00, u x is the mean of all possible observed values of x. 4.10 a. x 1 2 p ( x ) 9 4 9 4 9 1 b. x 1 2 3 p ( x ) 8 1 8 3 8 3 8 1 c. x 1 2 3 4 p ( x ) 16 1 4 1 8 3 4 1 16 1 4.11 a. [ x 2 x ] = [.667 2(.667)] = [.667, 2.001] contains at least 3 4 of the observed values of x . [ x 3 x ] = [.667 3(.667)] = [1.334, 2.668] contains at least 8 9 of the observed values of x . b. See the methods outlined in part a. x = 1.5 x 2 = .75 x = .866 [ x 2 x ] = [.232, 3.232] [ x 3 x ] = [1.098, 4.098] 44 Chapter 4: Discrete Random Variables c. See the methods outlined in part (a). x = 2 x 2 = 1 x = 1 [ x 2 x ] = [0, 4] [ x 3 x ] = [1, 5] 4.12 a. x 1 2 3 4 5 p ( x ) 15 1 15 2 5 1 15 4 3 1 b. p ( x ) 0 for each value of x . p ( x ) all x = 1 15 + 2 15 + 1 5 + 4 15 + 1 3 = 1 c. d. 4.13 a. Graph not included in this manual. b. x = (- 40,000)(.25) + (10,000)(.7) + (70,000)(.05) = $500 If numerous oil wells were dug, the average profit would be $500....
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Chap004_solutions - CHAPTER 4Discrete Random Variables 4.1...

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