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EE562a_HW_3 - Homework 3 Due Monday Work all 12 problems...

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EE 562a Homework 3 Due Monday June 19, 2006 Work all 12 problems . Problem 1. Suppose X is a mean-zero random vector with covariance matrix K X = 1 4 3 4 20 22 3 22 36 . Find a transformation A such that Y = AX has covariance matrix K Y = I . Problem 2. Let Z ( n ) be an i.i.d. Bernoulli sequence where P ( Z ( n ) = 1) = p, P ( Z ( n ) = 1) = q = 1 p. Let X ( n ) = n k =0 Z ( k ) where we take Z (0) = 0 = X (0). Then X ( n ) is a discrete random walk. Find R X ( n, m ) for this random walk. 1
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Problem 3. Let X n , n 1, denote a sequence of independent and identi- cally distributed zero-mean unit-variance Gaussian random variables. Define for n 2 Y n = 1 2 ( X n + X n - 1 ) . a. Does this sequence converge in the mean square sense and, if so, to what limit? b. Does this sequence converge in distribution and, if so, to what distri- bution? Problem 4. Suppose the discrete random variable U takes on values u = 0 , u = 1 2 , u = 1 each with probability 1/3. Determine whether each of the following sequences
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