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Unformatted text preview: E [cos( X 3 )], a quantity that is di±cult to compute analytically. Problem 2: Two random variables X and Y have joint density p X,Y ( x,y ) = b Ke-2 x 2 + y 2 2 xy ≥ xy < (a) Find K . (b) Show that X and Y are each Gaussian random variables. (c) Express the probability P [ X 2 + X > 2] in terms of the Q function. (d) Are X and Y jointly Gaussian? (e) Are X and Y independent? (f) Are X and Y uncorrelated? (g) Find the conditional density p X | Y ( x | y ). Is it Gaussian? Problem 3: The random vector X = ( X 1 X 2 ) T is Gaussian with mean vector m = (-2 , 1) T and covariance matrix K given by K = p 9-2-2 4 P (a) Let Y 1 = X 1 + 2 X 2 , Y 2 =-X 1 + X 2 . Find cov ( Y 1 ,Y 2 ). (b) Write down the joint density of Y 1 and Y 2 . (c) Express the probability P [ Y 1 > 2 Y 2 + 1] in terms of the Q function. Problems 4-6: Problems 3.1, 3.3, 3.5...
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This note was uploaded on 12/29/2011 for the course ECE 253 taught by Professor Brewer,f during the Fall '08 term at UCSB.
- Fall '08