problem set7 - ELEC 531 STATISTICAL SIGNAL PROCESSING...

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ELEC 531: STATISTICAL SIGNAL PROCESSING Department of Electrical and Computer Engineering Rice University Fall 2007 Problem Set 7 Due: November 6, 2007 Problem 7.1 Consider the joint density of the random variables X and Y shown in Fig- ure 1 and the hypothesis test H 0 : X 0 H 1 : X > 0 . The idea here is that we want to make a decision about the sign of the realization X = x based off the corresponding observation Y = y . 2 2 -2 -2 -1 -1 1 1 p X,Y ( x , y ) x y Figure 1: (a) Determine the prior probabilities π 0 and π 1 and make fully labeled sketches of the conditional densities p Y | H 0 ( y | H 0 ) and p Y | H 1 ( y | H 1 ). Assume the shaded region in the Figure is constant while all other values of the joint density are zero. (b) What is the decision rule that minimizes the probability of error? What is the resulting probability of error? (c) Sketch the receiver operating characteristic for this likelihood ratio test. Problem 7.2 In the binary communications system shown below, messages m = 0 and 1
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m = 1 occur with a priori probabilities 1 4 and 3 4 . When transmitted, these messages are
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