ps3 - UCSB Spring 2010 ECE 146B: Problem Set 3 Assigned:...

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Unformatted text preview: UCSB Spring 2010 ECE 146B: Problem Set 3 Assigned: April 20 Due: April 27 (by noon, in course homework box) Reading: Classnotes, Chapter 7 Topics: Gaussian models; Hypothesis testing Remark: In view of our maniacal focus on Gaussian noise for the rest of the course, I have included a number of non-Gaussian hypothesis testing problems in this homework, so you realize that the framework applies in great generality. Problem 1: The random vector X = ( X 1 X 2 ) T is Gaussian with mean vector m = (- 3 , 2) T and covariance matrix C given by C = parenleftbigg 4- 2- 2 9 parenrightbigg (a) Let Y 1 = 2 X 1- X 2 , Y 2 =- X 1 + 3 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 2 > 2 Y 1- 1] in terms of the Q function with positive arguments. (d) Express the probability P [ Y 2 1 > 3 Y 1 + 10] in terms of the Q function with positive arguments....
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