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# ps1 - EE 805 Random Processes and Linear Systems Jan 6 2012...

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Unformatted text preview: EE 805, Random Processes and Linear Systems Jan. 6, 2012 OSU, Winter 2012 Due: Jan. 18, 2012 Problem Set 1 Problem 1 Let = (0 , 1] , let F be a -algebra of subsets of which includes the intervals, and let P be a probability measure on F so that P ([ a,b ]) = b- a for 0 a b 1 . Consider the following sequence of random variables ( X n ) n 1 over this probability space: X n ( ) = n 2 , 1 /n, , 1 /n &amp;lt; 1 . (a) Find and sketch the CDF of X n , i.e., F X n ( x ) , for a generic n. (b) Does your sequence of distributions F X n ( x ) in part (a) converge to a distribution F X ( x )? If yes, plot the limiting cdf F X ( x ) . (c) Does ( X n ) n converge almost surely ? Justify your answer. (d) Does ( X n ) n converge in mean square sense ? Justify your answer. Problem 2 Let X n = X + (1 /n ) where P ( X = i ) = 1 / 6 , for i = 1 , 2 , , 6 , and let F n ( x ) denote the distribution function of X n ....
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