HW3 - ACM/EE 116 - Fall 2009 - Homework 3 Solution Handed...

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ACM/EE 116 - Fall 2009 - Homework 3 Solution Handed out: Nov 3, 2009, Due: Nov 17, 2009 (in class). Please write down your solutions clearly and concisely, box your an- swers, put problems in order and bind pages. Grader: Molei Tao 1. A function φ ( x ) is said to belong to the ‘upper class’ if P ( S n > φ ( n ) n i.o.) = 0. A consequence of the law of the iterated logarithm is that α log log x is in the upper class for all α > 2. Use the first Borel- Cantelli lemma to prove the much weaker fact that φ ( x ) = α log x is in the upper class for all α > 2, in the special case when the X i are independent N (0 , 1) variables. (Hint: recall that S n = n i =1 X i . Only need to consider φ ( x ) evaluated at integer x .) 2. Let Y be uniformly distributed on [ - 1 , 1] and let X = Y 2 . (a) Find the best predictor of X given Y, and of Y given X. (b) Find the best linear predictor of X given Y, and of Y given X. (Hint: a best predictor of A given B is a function f ( B ) that minimizes E ( A - f ( B )) 2 . Different function spaces to which f ( · ) is restricted may result in different best predictors.) 3. (a) Suppose that X 1 ,X 2 ,... is a sequence of random variables, each having a normal distribution, and such that X n D -→ X . Show that X has a normal distribution, possibly degenerate.
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HW3 - ACM/EE 116 - Fall 2009 - Homework 3 Solution Handed...

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