sec_5_2_and_5_3_sol

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

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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, 0 x 1 , 0 y 1 . So, E [ | X - Y | ] = Z 1 0 Z 1 0 | 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 0 R 1 0 | x - y | · 6 x 2 y dy dx = R 1 0 R x 0 ( x - y ) · 6 x 2 y dy dx + R 1 0 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

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Similarly, by the symmetry of the problem, f Y ( y ) = Z 30 20 3 380 , 000 ( x 2 + y 2 ) dy = 3 380 , 000 10 y 2 + 19 , 000 3 , 20 y 30 and E [ Y ] = 1 , 925 76 . So, Cov ( X, Y ) = 24 , 375 38 - 1 , 925 76 · 1 , 925 76 = - 625 5 , 776 ≈ - 0 . 108 b. Corr ( X, Y ) = Cov ( X, Y ) q V ar ( X ) V ar ( Y ) E [ X 2 ] = Z 30 20 x 2 · 3 380 , 000 10 x 2 + 19 , 000 3 dx = 37 , 040 57 So, V ar ( X ) = E [ X 2 ] - ( E [ X ]) 2 = 37 , 040 57 - 1 , 925 76 2 = 143 , 285 17 , 328 8 . 269 .
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