problem2_39 - STA 6934 Alla Revenko Problem 2.39 a) Let 1 2...

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Unformatted text preview: STA 6934 Alla Revenko Problem 2.39 a) Let 1 2 , X X be i.i.d. (0,1) N . Prove 1 2 X Y X = is (0,1) C , i.e. ( 29 2 1 1 Y x π + : In fact, 1 2 1 2 2 , ( ) ( , ) x x x x x f x f x y dy ∞-∞ = ∫ , Consider a bijection 2 2 : h → , s.t. ( 29 1 2 , ( , ) h x x x y = with 1 2 2 , x x y x x = = . Its inverse function 2 2 : h- → is ( 29 1 2 ( , ) , h x y x x- = with 1 2 , x xy x y = = , 1 2 2 1 2 , , ( , ) ( , ) x x x x x f x y f xy y J = g , where Jacobean 1 y x J y = = . Since 1, 2 1 2 2 2 1 ( , ) ( ) ( ) exp 2 2 x x x x x y f x y f x f y π + = =- we get ( 29 1 2 2 1 2 2 2 2 2 , , 1 1 1 ( , ) ( , ) exp exp 2 2 2 2 x x x x x x y x y f x y f xy y J y y π π + + = =- = ⋅- g g . Therefore, ( 29 1 2 1 2 2 2 2 , 1 1 ( ) ( , ) exp 2 2 x x x x x x y f x f x y dy y dy π ∞ ∞-∞-∞ + = = ⋅- ∫ ∫ Using the substitution: ( 29 ( 29 2 2 2 1 , 2 1 , x y t x...
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This note was uploaded on 01/15/2012 for the course STA 6934 taught by Professor Young during the Fall '08 term at University of Florida.

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problem2_39 - STA 6934 Alla Revenko Problem 2.39 a) Let 1 2...

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