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Homework 7 - EE 351K PROBABILITY RANDOM PROCESSES FALL 2011...

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EE 351K PROBABILITY & RANDOM PROCESSES FALL 2011 Instructor: Sujay Sanghavi [email protected] Homework 7 Due: October 27th in class Focus: Limit Theorems, MAP Rule (Section 5.1-5.4, 8.1-8.2 in Textbook) Problem 1 Let X 1 , X 2 , . . . be independent random variables that are uniformly distributed over [ - 1 . 1] . For each of the following cases, find whether Y 1 , Y 2 , . . . convergence to some limit in probability. (a) Y n = X n /n . (b) Y n = ( X n ) n . (c) Y n = min { X 1 , ..., X n } . (d) Y n = max { X 1 , ..., X n } . Problem 2 Suppose that X 1 , X 2 , . . . are independent random variables, each with the exponential distribution with parameter λ = 1 . For n 2 , let Z n = max { X 1 ,...,X n } ln n . (a) Find a simple expression for the CDF of Z n . (b) Find the limit of this CDF. Is this limit a well-defined CDF? If not, regardless of all non-continuous points, what should the limit CDF be? (c) Show { Z n } n 2 converge in probability based on the result from part (b). Problem 3 Let X i , i = 1 , 2 , . . . , n be i.i.d. continuous random variables that are uniformly distributed between [0 , 1] .
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