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hw10_soln - Probability and Stochastic Processes A Friendly...

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Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J. Goodman Problem Solutions : Yates and Goodman,5.5.3 5.7.5 5.8.2 5.8.3 5.9.1 5.9.2 5.7.6 and 5.9.3 Problem 5.5.3 The joint PDF of X and Y and the region of nonzero probability are f X Y x y 5 x 2 2 1 x 1 0 y x 2 0 otherwise (a) The first moment of X is E X 1 1 x 2 0 x 5 x 2 2 dydx 1 1 5 x 5 2 dx 5 x 6 12 1 1 0 Since E X 0, the variance of X and the second moment are both Var X E X 2 1 1 x 2 0 x 2 5 x 2 2 dydx 5 x 7 14 1 1 10 14 (b) The first and second moments of Y are E Y 1 1 x 2 0 y 5 x 2 2 dydx 5 14 E Y 2 1 1 0 x 2 y 2 5 x 2 2 dydx 5 26 Therefore, Y has variance Var Y 5 26 5 14 2 0576 (c) Since E X 0, Cov X Y E XY E X E Y E XY . Thus, Cov X Y E XY 1 1 x 2 0 xy 5 x 2 2 dydx 1 1 5 x 7 4 dx 0 (d) The expected value of the sum X Y is E X Y E X E Y 5 14 (e) By Theorem 5.10, the variance of X Y is Var X Y Var X Var Y 2Cov X Y 5 7 0 0576 0 7719 1
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Problem 5.7.5 Random variables X and Y have joint PDF f X Y x y 1 2 1 x y 1 0 otherwise (a) For 1 y 1, the marginal PDF of Y is f Y y f X Y x y dx 1 2 y 1 dx y 1 2
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