Dr. Hackney STA Solutions pg 19

Dr. Hackney STA Solutions pg 19 - 2-4 Solutions Manual for...

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2-4 Solutions Manual for Statistical Inference 2.9 From the probability integral transformation, Theorem 2.1.10, we know that if u ( x ) = F x ( x ), then F x ( X ) uniform(0 , 1). Therefore, for the given pdf, calculate u ( x ) = F x ( x ) = 0 if x 1 ( x - 1) 2 / 4 if 1 < x < 3 1 if 3 x . 2.10 a. We prove part b), which is equivalent to part a). b. Let A y = { x : F x ( x ) y } . Since F x is nondecreasing, A y is a half infinite interval, either open, say ( -∞ , x y ), or closed, say ( -∞ , x y ]. If A y is closed, then F Y ( y ) = P ( Y y ) = P ( F x ( X ) y ) = P ( X A y ) = F x ( x y ) y. The last inequality is true because x y A y , and F x ( x ) y for every x A y . If A y is open, then F Y ( y ) = P ( Y y ) = P ( F x ( X ) y ) = P ( X A y ) , as before. But now we have P ( X A y ) = P ( X ( - ∞ ,x y )) = lim x y P ( X ( -∞ , x ]) , Use the Axiom of Continuity, Exercise 1.12, and this equals lim x y F X ( x ) y . The last inequality is true since F x ( x ) y for every x A y , that is, for every x < x y . Thus, F Y ( y ) y for every y . To get strict inequality for some y , let y be a value that is “jumped over” by F x . That is, let y be such that, for some x y , lim x y F X ( x ) < y < F X ( x y ) . For such a y , A y = (
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