HW6-solution

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

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Unformatted text preview: 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 /σ X ) = 0 . 55. Using Table 3.1 for the Gaussian CDF, we find that 10 /σ X = . 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 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 /σ X ) = 2 . 3 · 10- 4 (2) From Table 3.2, this implies 14 /σ X = 3 . 5 or σ X = 4. 1 (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 . × 10- 11 . (c) First we need to find the probability that a man is at least 7’6”....
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This note was uploaded on 08/25/2008 for the course EEE 350 taught by Professor Duman during the Spring '08 term at ASU.

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

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