HW8Sol(1) - Econ 3190 Fall 2011 Solution to HW8 1. f ( x) =...

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Econ 3190 Fall 2011 Solution to HW8 1. f ( x ) = Z 1 x 2 dy = 2(1 x ) for 0 < x < 1 f ( x ) = 0 elsewhere EX = Z 1 0 x 2(1 x ) dx = 1 3 EX 2 = Z 1 0 x 2 2(1 x ) dx = 1 6 2 x = EX 2 ( EX ) 2 = 1 18 f ( y ) = Z y 0 2 dx = 2 y for 0 < y < 1 f ( y ) = 0 elsewhere EY = Z 1 0 2 y 2 dy = 2 3 EY 2 = Z 1 0 2 y 3 dy = 1 2 2 y = EY 2 ( EY ) 2 = 1 18 EXY = Z 1 0 Z y 0 2 xydxdy = 1 4 ± = cov ( X;Y ) x y = 1 4 1 3 2 3 1 18 = 1 2 2. V ar ( Z ) = 4 V ar ( X ) + 16 V ar ( Y ) = 4 ± 5 + 16 ± 3 = 68 1
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3. f ( x ) = Z 1 0 3 2 ( x 2 + y 2 ) dy = 3 2 ( x 2 y + 1 3 y 3 ) j 1 0 = 3 2 x 2 + 1 2 for 0 < x < 1 f ( x ) = 0 elsewhere f ( y ) = Z 1 0 3 2 ( x 2 + y 2 ) dx = 3 2 y 2 + 1 2 for 0 < y < 1 f ( y ) = 0 elsewhere we observe f ( x;y ) 6 = f ( x ) f ( y ) ; that is to say X and Y are not independent. EX = Z 1 0 x ( 3 2 x 2 + 1 2 ) dx = 5 8 EY = 5 8 = : 625 E ( X + Y ) = EX + EY = 5 4 EXY = Z 1 0 Z 1 0 xy 3 2 ( x 2 + y 2 ) dxdy = 3 8 = : 375 2
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EX 2 = Z 1 0 x 2 ( 3 2 x 2 + 1 2 ) dx = 7 15 = 0 : 47 EY 2 = 7 15 V ar ( X ) = EX
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HW8Sol(1) - Econ 3190 Fall 2011 Solution to HW8 1. f ( x) =...

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