hw04_sol - Homework 4 Problem 1 Notice that R 1 f x dx = 1...

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Unformatted text preview: Homework 4 September 21, 2007 Problem 1. Notice that R 1 f ( x ) dx = 1 and R 1 xf ( x ) dx = c . Then Z 1 ( a + bx 2 ) dx = 1 , Z 1 x ( a + bx 2 ) dx = c. Arranging these equations, we have a + b 3 = 1 , a 2 + b 4 = c. Hence, a =- 4 c + 3 and b = 12 c- 6 . Problem 2. f x = 6 7 R 2 ( x 2 + xy 2 ) dy = 6 7 ( x 2 y | 2 + xy 2 4 | 2 ) = 6 7 (2 x 2 + x ) f y = 6 7 R 1 ( x 2 + xy 2 ) dx = 6 7 ( x 3 3 | 1 + x 2 y 4 | 1 ) = 6 7 ( 1 3 + y 4 ) μ = E [ X ] = R 1 f x xdx = 6 7 R 1 (2 x 3 + x 2 ) dx = 6 7 ( 2 x 4 4 | 1 + x 3 3 | 1 ) = 5 7 ν = E [ Y ] = R 2 f y ydy = 6 7 R 2 ( y 3 + y 2 4 ) dy = 6 7 ( y 2 6 | 2 + y 3 12 | 2 ) = 8 7 Cov ( X,Y ) = E [( X- μ )( Y- ν )] = E [ X · Y ]- μν = R R x · y · f ( x,y ) dxdy- μν = 6 7 R 2 R 1 ( x 3 y + x 2 y 2 2 ) dxdy- μν = 6 7 R 2 ( yx 4 4 + x 3 y 2 6 ) | 1 dy- μν = 6 7 R 2 ( y 4 + y 2 6 ) dy- μν = 6 7 ( y 2 8 + y 3 18 ) | 2- 5 7 · 8 7 = 17 21- 40 49 =- 1 147 Problem 3....
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This note was uploaded on 11/12/2008 for the course ORIE 360 taught by Professor Ehrlichman during the Fall '07 term at Cornell.

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hw04_sol - Homework 4 Problem 1 Notice that R 1 f x dx = 1...

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