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Unformatted text preview: Z . 2 5. (a) X and Y are independent and identically distributed normal random variables with mean 0 and variance . 2 σ . 2 1 ) ( ) ( ) , ( 2 2 2 2 / ) ( 2 πσ y x Y X XY e y f x f y x f +-= = Define . 2 , 2 2 2 2 2 2 Y X XY V Y X Y X U + = +-= Find the joint p.d.f ) , ( v u f UV of the random variables U and V . Using the expression for ) , ( v u f UV conclude whether U and V are independent or not. (b) X and Y are independent exponential r.vs with common parameter . λ Find (i) [ ] ) , min( Y X E (ii) [ ] . ) , 2 max( Y X E ( Hint: Use the fact that [ ] ∫∫ = dxdy y x f y x g Y X g E XY ) , ( ) , ( ) , ( )....
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- Spring '11
- Variance, Probability theory