pracprelim_soln

pracprelim_soln - ORIE 3500/5500, Fall ’10 Solutions for...

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Unformatted text preview: ORIE 3500/5500, Fall ’10 Solutions for Practice Problems Solutions for Selected Practice Problems Problem 2 (a) The marginal densities of X and Y are given by f X ( x ) = Z ∞-∞ f X,Y ( x,y ) dy = Z 2 1 10 (4 x + 2 y + 1) dy = 1 10 (4 xy + y 2 + y ) y =2 y =0 = 1 10 (8 x + 6) = 1 5 (4 x + 3) , < x < 1 and f Y ( y ) = Z ∞-∞ f X,Y ( x,y ) dx = Z 1 1 10 (4 x + 2 y + 1) dx = 1 10 (2 x 2 + 2 xy + x ) x =1 x =0 = 1 10 (2 y + 3) , < y < 2 . Therefore, E ( X ) = Z ∞-∞ xf X ( x ) dx = Z 1 1 5 (4 x 2 + 3 x ) dx = 1 5 4 3 x 3 + 3 2 x 2 1 = 17 30 and E ( Y ) = Z ∞-∞ yf Y ( y ) dy = Z 2 1 10 (2 y 2 + 3 y ) dy = 1 10 2 3 y 3 + 3 2 y 2 2 = 17 15 . Also, E ( XY ) = Z ∞-∞ Z ∞-∞ xyf X,Y ( x,y ) dxdy = Z 2 Z 1 1 10 (4 x 2 y + 2 xy 2 + xy ) dxdy = Z 2 1 10 4 3 x 3 y + x 2 y 2 + 1 2 x 2 y 1 dy = Z 2 1 10 y 2 + 11 6 y dy = 1 10 1 3 y 3 + 11 12 y 2 2 = 19 30 . 1 ORIE 3500/5500, Fall ’10 Solutions for Practice Problems Therefore, Cov( X,Y ) = E ( XY )- E ( X ) E ( Y ) = 19 30- 17 30...
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This note was uploaded on 10/29/2011 for the course MATH 3310 at Cornell.

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pracprelim_soln - ORIE 3500/5500, Fall ’10 Solutions for...

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