sec_5_2_and_5_3_sol

sec_5_2_and_5_3_sol - APPM 4/5570 Solutions to Problems...

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Unformatted text preview: APPM 4/5570 Solutions to Problems from Section 5.2 and 5.3 27. The waiting time is either X- Y or Y- X , depending on who arrives first. We can thus write the waiting time as | X- Y | . E [ | X- Y | ] = Z ∞-∞ Z ∞-∞ | x- y | f X,Y ( x, y ) dy dx. Since X and Y are independent, f X,Y ( x, y ) = f X ( x ) · F Y ( y ) = 6 x 2 y, ≤ x ≤ 1 , ≤ y ≤ 1 . So, E [ | X- Y | ] = Z 1 Z 1 | x- y | · 6 x 2 y dy dx Note that | x- y | = x- y , if x ≥ y y- x , if x ≤ y So, E [ | X- Y | ] = R 1 R 1 | x- y | · 6 x 2 y dy dx = R 1 R x ( x- y ) · 6 x 2 y dy dx + R 1 R 1 x ( y- x ) · 6 x 2 y dy dx = 1 / 6 + 1 / 12 = 3 / 12 = 1 / 4 . 31. a. Cov ( X, Y ) = E [ XY ]- E [ X ] E [ Y ] E [ XY ] = R 30 20 xy · 3 380 , 000 ( x 2 + y 2 ) dydx = 24 , 375 38 ≈ 641 . 45 f X ( x ) = Z 30 20 3 380 , 000 ( x 2 + y 2 ) dy = 3 380 , 000 10 x 2 + 19 , 000 3 , 20 ≤ x ≤ 30 . So, E [ X ] = Z 30 20 x · 3 380 , 000 10 x 2 + 19 , 000 3 dx = 1 , 925 76 Similarly, by the symmetry of the problem, f Y ( y...
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This note was uploaded on 01/23/2011 for the course APPM 5440 at Colorado.

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sec_5_2_and_5_3_sol - APPM 4/5570 Solutions to Problems...

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