Hw8sol - ρ ( X, Y ) = 0 . 8. 5. Cov( X, Y ) = E[( X-E[ X...

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ISyE 2027B Probability with Applications Fall 2010 Homework 8 Solutions 1. Cov( U, V ) = E[( U - E[ U ])( V - E[ V ])] = E[ UV ] - E[ U ]E[ V ] = 0 × 1 4 + 1 × 1 2 + 0 × 1 4 - (0 × 1 4 + 1 × 1 2 + 2 × 1 4 )(0 × 1 2 + 1 × 1 2 ) = 1 2 - 1 2 = 0 ρ ( U, V ) = 0 2. Similar to Problem 1, compute marginal probabilities first. Both X and Y have marginal probabilities (1 / 4 , 1 / 4 , 1 / 4 , 1 / 4). E[ X ] = 1 × 1 4 + 2 × 1 4 + 3 × 1 4 + 4 × 1 4 = 2 . 5 E[ Y ] = 1 × 1 4 + 2 × 1 4 + 3 × 1 4 + 4 × 1 4 = 2 . 5 E[ XY ] = 1 × 1 × 16 136 + . . . + 4 × 4 × 1 136 = 6 . 25 Cov( X, Y ) = E[ XY ] - E[ X ]E[ Y ] = 6 . 25 - 2 . 5 × 2 . 5 = 0 3. (a) E[ X 2 ] = V ar ( X ) + (E[ X ]) 2 = 4 + 2 2 = 8. (b) E[ - 2 X 2 + Y ] = - 2E[ X 2 ] + E[ Y ] = - 2 × 8 + 3 = - 13. 4. For constant a, b, c, d R , Cov( aX + b, cY + d ) = ab Cov( X, Y ), ρ ( aX + b, cY + d ) = ρ ( X, Y ). Since T = 9 / 5 X + 32, S = 9 / 5 Y + 32. Thus Cov( T, S ) = (9 / 5) 2 Cov( X, Y ) = 9 . 72 and ρ ( T, S ) =
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Unformatted text preview: ρ ( X, Y ) = 0 . 8. 5. Cov( X, Y ) = E[( X-E[ X ])( Y-E[ Y ])] = E[ XY-X E[ Y ]-Y E[ X ] + E[ X ]E[ Y ]] = E[ XY ]-E[ X ]E[ Y ]-E[ X ]E[ Y ] + E[ X ]E[ Y ] = E[ XY ]-E[ X ]E[ Y ] 6. (a) Since ∫ ∞-∞ ∫ ∞-∞ f ( x, y ) dxdy = ∫ 1 ∫ 1 kxdxdy = k/ 2 = 1, we take k = 2. (b) E[ X ] = ± 1 ± 1 2 x 2 dxdy = 2 3 E[ Y ] = ± 1 ± 1 2 xydxdy == 1 2 E[ XY ] = ± 1 ± 1 2 x 2 ydxdy = 1 3 Cov( X, Y ) = E[ XY ]-E[ X ]E[ Y ] = 0 1...
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This note was uploaded on 02/21/2011 for the course ISYE 2027 taught by Professor Zahrn during the Fall '08 term at Georgia Institute of Technology.

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