problem2_39

# problem2_39 - STA 6934 Alla Revenko Problem 2.39 a Let X 1...

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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 0 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 y dy dt + = + = we get ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 2 2 2 0 0 0 0 2 2 1 1 ( ) 2 2 1 1 1 1 1 1 1 t t t x x e f x dt e dt e x x x e e x x π π π π π + ∞ + ∞ - + ∞ - - -∞ - = = = - = + + + = - + = + + Thus, we proved that ( 29 2 1 1 Y x π + : , i.e. Y is Cauchy.

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b) In this part of the problem 2.39 we have to show that the Cauchy distribution
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