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HW6-solution

HW6-solution - Probability and Stochastic Processes A...

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Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Edition 2 Roy D. Yates and David J. Goodman Problem Solutions : Yates and Goodman, 3.5.3 3.5.4 3.5.7 3.5.8 3.5.10 3.7.2 3.7.5 3.7.7 3.7.11 and 3.7.16 Problem 3.5.3 Solution X is a Gaussian random variable with zero mean but unknown variance. We do know, however, that P [ | X | ≤ 10] = 0 . 1 (1) We can find the variance Var[ X ] by expanding the above probability in terms of the Φ( · ) function. P [ - 10 X 10] = F X (10) - F X ( - 10) = 2Φ 10 σ X - 1 (2) This implies Φ(10 X ) = 0 . 55. Using Table 3.1 for the Gaussian CDF, we find that 10 X = 0 . 15 or σ X = 66 . 6. Problem 3.5.4 Solution Repeating Definition 3.11, Q ( z ) = 1 2 π Z z e - u 2 / 2 du (1) Making the substitution x = u/ 2, we have Q ( z ) = 1 π Z z/ 2 e - x 2 dx = 1 2 erfc z 2 (2) Problem 3.5.7 Solution We are given that there are 100 , 000 , 000 men in the United States and 23 , 000 of them are at least 7 feet tall, and the heights of U.S men are independent Gaussian random variables with mean 5 0 10 00 . (a) Let H denote the height in inches of a U.S male. To find σ X , we look at the fact that the probability that P [ H 84] is the number of men who are at least 7 feet tall divided by the total number of men (the frequency interpretation of probability). Since we measure H in inches, we have P [ H 84] = 23 , 000 100 , 000 , 000 = Φ 70 - 84 σ X = 0 . 00023 (1) Since Φ( - x ) = 1 - Φ( x ) = Q ( x ), Q (14 X ) = 2 . 3 · 10 - 4 (2) From Table 3.2, this implies 14 X = 3 . 5 or σ X = 4. 1
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(b) The probability that a randomly chosen man is at least 8 feet tall is P [ H 96] = Q 96 - 70 4 = Q (6 . 5) (3) Unfortunately, Table 3.2 doesn’t include Q (6 . 5), although it should be apparent that the probability is very small. In fact, Q (6 . 5) = 4 . 0 × 10 - 11 . (c) First we need to find the probability that a man is at least 7’6”. P [ H 90] = Q 90 - 70 4 = Q (5) 3 · 10 - 7 = β (4) Although Table 3.2 stops at Q (4 . 99), if you’re curious, the exact value is Q (5) = 2 . 87 · 10 - 7 .
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