Solution to HW 9 - ) var( ) 2cov( , ) 162 81 81 162 X Y X Y...

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1. a. The marginal distribution of X is 2 , 0 2 ( ) ( , ) ( 1) 3 3 X X Y x y f x f x y dy dy x +∞ -∞ + = = = + , where 0 1 x ≤ ≤ , hence 1 3 2 1 0 0 2 2 5 ( ) ( ) ( 1) ( ) | 3 3 3 2 9 X x x E X xf x dx x x dx +∞ -∞ = == + = + = b. The marginal distribution of Y is 1 , 0 1 1 ( ) ( , ) 3 3 6 Y X Y x y f y f x y dx dx y +∞ -∞ + = = = + , where 0 2 y ≤ ≤ , hence 2 3 2 2 0 0 1 1 11 ( ) ( ) ( ) ( ) | 3 6 9 12 9 Y y y E Y yf y dy y y dy +∞ -∞ = == + = + = c. 1 4 3 2 2 2 1 0 0 2 2 7 ( ) ( ) ( 1) ( ) | 3 3 4 3 18 X x x E X x f x dx x x dx +∞ -∞ = == + = + = 2 2 2 7 5 13 var( ) ( ) ( ) ( ) 18 9 162 X E X E X = - = - = d. 2 4 3 2 2 2 2 0 0 1 1 16 ( ) ( ) ( ) ( ) | 3 6 12 18 9 Y y y E Y y f y dy y y dy +∞ -∞ = == + = + = 2 2 2 16 11 23 var( ) ( ) ( ) ( ) 9 9 81 Y E Y E Y = - = - = e. 1 2 , 0 0 ( ) ( , ) 3 X Y x y E XY xyf x y dydx xy dydx +∞ +∞ -∞ -∞ + = = ∫ ∫ ∫∫ 1 2 1 2 0 0 0 2 8 2 ( ) 3 3 9 3 x y xy dydx x x dx + = = + = ∫∫ 2 5 11 1 cov( , ) ( ) ( ) ( ) 3 9 9 81 X Y E XY E X E Y = - = - = - According to page 239 in the textbook, 13 23 2 55 var( ) var(
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Unformatted text preview: ) var( ) 2cov( , ) 162 81 81 162 X Y X Y X Y + = + + = +-= 2. 2 2 2 2 ( , ) ( ) ( ) ( ) [ ( )] ( ) ( ) [ ] [ ( )] ( ) ( ) [ ( )] ( ) Cov X Y E XY E X E Y E X aX b E X E aX b E aX bX a E X bE X aE X a E X aVar X =-= +-+ = +--=-= 2 ( ) ( ) ( ) ( ) ( ) ( ) | | ( ) Var X Var Y Var X Var aX b Var X a Var X a Var X = + = = Hence 1 ( , ) ( ) ( , ) 1 | | ( ) ( ) ( ) a Cov X Y aVar X X Y a a Var X Var X Var Y > = = = -< 3....
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Solution to HW 9 - ) var( ) 2cov( , ) 162 81 81 162 X Y X Y...

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