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20105ee131A_1_ps7_2010fall_CW_sol

# 20105ee131A_1_ps7_2010fall_CW_sol - EE 131A Problem Set 7...

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Unformatted text preview: EE 131A Problem Set 7 Probability Wednesday, November 17, 2010 Instructor: Lara Dolecek Due: Wednesday, November 24, 2010 Happy Thanksgiving! Reading: Chapters 5 and 6 of Probability, Statistics, and Random Processes by A. Leon-Garcia 100 points total 1. Expectation of a function of RVs. Problem 5.57, page 295 of ALG The expectation is shown as the following: E ( | X- Y | ) = R ∞ R ∞ | x- y | f XY ( x,y ) dxdy = R ∞ x =0 R x y =0 ( x- y ) f XY ( x,y ) dydx + R ∞ y =0 R y x =0 ( y- x ) f XY ( x,y ) dxdy = R ∞ x =0 xe- x (1- e- 2 x ) + e- x ( xe- 2 x + 1 2 e- 2 x- 1 2 ) dx + R ∞ y =0 2 ye- 2 y (1- e- y ) + 2 e- 2 y ( ye- y + e- y- 1) dy = R ∞ x =0 xe- x + 1 2 e- 3 x- 1 2 e- x dx + R ∞ y =0 2 ye- 2 y + 2 e- 3 y- 2 e- 2 y dy = (1 + 1 6- 1 2 ) + ( 1 2 + 2 3- 1) = 5 6 2. Joint cdf, pdf refresher. Problem 5.27, page 292 of ALG (a) The integration on f xy through the range of x and y should be one. R R f XY ( x,y ) = 1 = R 1 R 1 kx (1- x ) ydydx = 0 . 5 k R 1 x- x 2 dx = 1 12 k = 1 So we have k = 12. (b) The cdf is given as the following: F XY ( x,y ) = 12 Z x Z y f XY ( t,u ) dudt = (3 x 2- 2 x 3 ) y 2 1 (c) The marginal pdf’s are given as the following: f X ( x ) = R 1 f XY ( x,y ) dy = 6 x (1- x ) f Y ( y ) = R 1 f XY ( x,y ) dx = 2 y (d) The probabilities are given as the following: P ( Y < X . 5 ) = 12 R 1 R x . 5 f XY ( x,y ) dydx = 6 R 1 x (1- x ) xdx = 0 . 5 P ( X < Y ) = 12 R 1 R y f XY ( x,y ) dxdy = R 1 6 y 3- 4 y 4 dy = 0 . 7 3. Joint moments. Find the correlation and covariance of X and Y in the previous prob- lem. Are X and Y independent ? orthogonal ? uncorrelated?...
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20105ee131A_1_ps7_2010fall_CW_sol - EE 131A Problem Set 7...

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