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# hw5 - E | X | = ∞(This gives an example in which lim n X...

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Math 280A, Fall 2011 Homework 5 1. Problem 16 (page 116) 2. Problem 21 (page 117) 3. (a) Let U and V be independent random variables, with Var( U ) = Var( V ) < . Define X = U + V and Y = U V . Show that X and Y are uncorrelated; i.e. , that Cov( X, Y ) = 0. (b) Toss two fair dice. Let X be the sum of the numbers showing on the two dice, and let Y be the difference. Are X and Y uncorrelated? Are X and Y independent? 4. Let U and V be independent random variables, each with the standard normal distri- bution. Define X n := U | V | + 1 /n , n = 1 , 2 , . . .. (a) Use Fubini’s theorem to show that E [ X n ] = 0 for all n . (b) Observe that X := lim n X n = U/ | V | . Show that
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Unformatted text preview: E | X | = + ∞ . (This gives an example in which lim n X n = X but lim n E [ X n ] n = E [ X ]; indeed the latter expectation doesn’t even exist.) 5. Let A 1 , A 2 , . . ., A n , . . . be an infnite sequence oF events From a common probability space (Ω , F , P ). ²ix ǫ ∈ (0 , 1). (a) Show that iF P [ A n ] ≥ ǫ For all n , then P [ A n i.o. ] ≥ ǫ . (b) Show that iF A 1 , A 2 , . . ., A n , . . . are independent and P [ A n ] ≥ ǫ For all n , then P [ A n i.o. } = 1....
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