ps3 - Massachusetts Institute of Technology Department of...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.432 Stochastic Processes, Detection and Estimation Problem Set 3 Spring 2004 Issued: Thursday, February 19, 2004 Due: Thursday, February 26, 2004 Reading: This problem set: Chapter 2, Chapter 3 through Section 3.2.4 Next: Sections 1.7, 3.2.5, 3.3.1, 3.3.2 Problem 3.1 Suppose x and y are the random variables from Problem Set 2, problem 2.4. Their joint density, depicted again below for convenience, is constant in the shaded area x y p ( x , y ) 1 2 1 2 and 0 elsewhere. x,y -1 -2 -1 -2 (a) In the ( P D , P F ) plane sketch the operating characeristic of the likelihood ratio test for this problem. Also, indicate on this plot the region consisting of every ( P D , P F ) value that can be achieved using some decision rule. 2 (b) Is the point corresponding to P D = 3 , P F = 5 in this region? If so, describe a 6 test that achieves this value. If not, explain. 1
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Problem 3.2 We observe a random variable y and have two hypotheses, H 0 and H 1 , for its prob- ability density. In particular, the probability densities for y under each of these two hypotheses are depicted below: p y | H 0 ( y | H 0 ) = 1 , 0 y 1 p y | H 1 ( y | H 1 ) = 3 y 2 , 0 y 1 3 1 y y 1 1 (a) Find the decision rule that maximize P D subject to the constraint that P F 1 . (b) Determine the value of P D for the decision rule speci±ed in part (a). Problem 3.3 Let k denote the uptime of a communications link in days. Given that the link is functioning at the beginning of a particular day there is probability q that it will go down that day. Thus, the uptime of the link k (in days) obeys a geometric distribution. It
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ps3 - Massachusetts Institute of Technology Department of...

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