EE562a_HW_3

EE562a_HW_3 - Homework 3 Due Monday June 19, 2006 Work all...

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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 = 14 3 42 02 2 32 23 6 . 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 X 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. DeFne 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 =1 each with probability 1/3. Determine whether each of the following sequences of random variables converge i) surely, ii) almost surely, iii) in mean square or iv) none of these. If the sequence converges in one of these types specify
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This note was uploaded on 09/13/2008 for the course EE 562a taught by Professor Toddbrun during the Summer '07 term at USC.

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EE562a_HW_3 - Homework 3 Due Monday June 19, 2006 Work all...

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