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Unformatted text preview: Homework Set No. 5 – Probability Theory (235A), Fall 2011 Due: Tuesday 11/01/11 at discussion section 1. (a) If X ≥ 0 is a nonnegative r.v. with distribution function F , show that E ( X ) = Z ∞ P ( X ≥ x ) dx. (b) Prove that if X 1 ,X 2 ,..., is a sequence of independent and identically distributed (“i.i.d.”) r.v.’s, then P ( | X n | ≥ n i.o.) = 0 if E | X 1 | < ∞ , 1 if E | X 1 | = ∞ . (c) Deduce the following converse to the Strong Law of Large Numbers in the case of undefined expectations: If X 1 ,X 2 ,... are i.i.d. and E X 1 is undefined (meaning that E X 1+ = E X 1- = ∞ ) then P lim n →∞ 1 n n X k =1 X k does not exist ! = 1 . 2. Let X be a r.v. with finite variance, and define a function M ( t ) = E | X- t | , the “mean absolute deviation of X from t ”. The goal of this question is to show that the function M ( t ), like its easier to understand and better-behaved cousin, E ( X- t ) 2 (the “moment of inertia” around t , which by the Huygens-Steiner theorem is simply a parabola in t , taking its minimum value of V...
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- Fall '11