hw6 - 4. Let { A n } be a sequence of events with lim n P [...

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Math 280A, Fall 2011 Homework 6 1. Problem 2 (page 153) 2. Problem 5 (page 154) 3. In this problem let X and Y be independent random variables with cumulative distri- bution functions F and G ; thus, for example, F ( x ) = P [ X x ] for x R . (a) Use Tonelli’s theorem to show that E [ F ( Y )] = P [ X Y ] and E [ G ( X )] = P [ Y X ]. (b) Deduce from (a) that E [ F ( Y )] + E [ G ( X )] = 1 + P [ X = Y ]. (c) Use Tonelli’s theorem again to show that P [ X = Y ] = b Δ F ( b ) · Δ G ( b ), where Δ F ( b ) := F ( b ) - F ( b - ), G ( b ) := G ( b ) - G ( b - ), and the sum extends over the (at most countable) set of real numbers b with Δ F ( b ) · Δ G ( b ) > 0—the set of common discontinuities of F and G . (Thus P [ X = Y ] = 0 if and only if F and G have no common discontinuities.]
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Unformatted text preview: 4. Let { A n } be a sequence of events with lim n P [ A n ] = 0 and ∑ ∞ n =1 P [ A n \ A n +1 ] < ∞ . Show that P [ A n i.o.] = 0. [Hint: ∪ m k = n A k = ∪ m-1 k = n [ A k \ A k +1 ] ∪ A m .] 5. In this problem X is a non-negative random variable. (a) Suppose there are constants p > 0 and C > 0 such that P [ X > n ] ≤ C · n-p ,n = 1 , 2 ,.... Show that E [ X r ] < ∞ provided 0 < r < p . (b) Suppose that r > 0 and that E [ X r ] < ∞ . Show that lim x →∞ x r P [ X > x ] = 0 ....
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This note was uploaded on 12/11/2011 for the course MATH 280a taught by Professor Driver,b during the Fall '08 term at UCSD.

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