This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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”....
View
Full
Document
This note was uploaded on 08/25/2008 for the course EEE 350 taught by Professor Duman during the Spring '08 term at ASU.
 Spring '08
 Duman

Click to edit the document details