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Unformatted text preview: EE 805, Random Processes and Linear Systems Jan. 15, 2012 OSU, Winter 2012 Due: Jan. 25, 2012 Problem Set 2 Problem 1 Let C be a random variable uniformly distributed on [0 , 1] and define X ( t ) = u ( t C ). (a) Sketch two sample realizations of X ( t ). (b) Find F X ( t ) ( x ) and f X ( t ) ( x ) for all x and t . (c) Find E [ X ( t )]. Problem 2 Let A be a r.v. with exponential pdf f A ( a ) = 2 e 2 a u ( a ). Let X ( t ) = e At u ( t ). (a) Sketch a few sample realizations for X ( t ). (b) Find the CDF and pdf of X ( t ). Sketch them for t = 2 and t = 6. Do these sketches agree with your intuition based on plotting a few sample realizations? Problem 3 Consider the random process X [ n ] defined as X [ n ] = 1 2 X [ n 1] + W [ n ] , n > 1 , and X [0] = W [0], where samples W [ k ] are iid N (0 , 1) , k > 0. (a) Sketch a sample realization for X [ n ]. (b) Is it possible to find a 3 3 matrix A such that [ X [0] X [1] X [2]] T = A [ W [0] W [1] W [2]] T ? If so, find A ....
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 Spring '12
 Eryilmaz

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