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First Course in Probability, A (7th Edition)

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Fifth Homework Set — Solutions Chapter 4 Problem 17 (a) P { X = 1 } = P { X 1 } - P { X < 1 } = 1 2 - 1 4 = 1 4 P { X = 2 } = 11 12 - 3 4 = 1 6 P { X = 3 } = 1 - 11 12 = 1 12 (b) P braceleftbig 1 2 < X < 3 2 bracerightbig = 5 8 - 1 8 = 1 2 . Problem 19 P { X = 0 } = 1 2 P { X = 1 } = 3 5 - 1 2 = 1 10 P { X = 2 } = 4 5 - 3 5 = 1 5 P { X = 3 } = 9 10 - 4 5 = 1 10 P { X = 3 . 5 } = 1 - 9 10 = 1 10 Problem 21 (a) E [ X ] is larger than E [ Y ] because the random selection of students favors larger busloads. (b) E [ X ] = 40 · 40+33 · 33+25 · 25+50 · 50 40+33+25+50 = 5814 148 = 39 . 3, E [ Y ] = 148 4 = 37. Problem 23 (a) Suppose that that you use x dollars to buy x/ 2 ounces of the commodity and keep the rest 1000 - x dollars as cash, and then sell your commodity at the end of the week. Then the expected amount of money you have at the end of the week is 1 2 x 2 + 1 2 2 x + 1000 - x = 1000 + x 4 which is an increasing function of x . Therefore the best strategy is to use all your money to buy 500 ounces of the commodity and then sell at the end of the week. 1
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(b) Suppose that you use x dollars to buy x/ 2 ounces of the com- modity at the beginning of the first week and use the remaining 1000 - x
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