RP-HW5 - Kyung Hee University Department of Electronics and...

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1 Kyung Hee University Department of Electronics and Radio Engineering C1002900 Random Processing Homework 5 Spring 2010 Professor Hyundong Shin Issued: May 17, 2010 Due: May 31, 2010 (No acceptance of overdue submission) Reading: Course textbook Chapter 8, Lecture Note X HW 5.1 Suppose X and Y are random variables whose joint density, depicted in Fig. 5.1, is constant in the shaded area and 0 elsewhere. Figure 5.1 (a) Let H H 0 when X 0 , and H H 1 when X 0 . Determine  P HH  00 ± and P 11 . Make fully labeled sketches of   YH f yH 0 and   f 1 . (b) Design a rule  ˆ H y for deciding between H 0 and H 1 given an observation Y y , which minimizes the probability of error. Specify for which values of Y your rule chooses H 1 , and for which values it chooses H 0 . That is, determine the regions ˆ ˆ . yH y H What is the resulting probability of error? (c) Sketch the operating characteristic of the likelihood ratio test (LLT) for this problem in the   DF , P P plane. On this plot, indicate also the region consisting of every   , P P value that can be achieved using some decision rule.   , , XY fx y y x
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2 (d) Is the point corresponding to D P 23 and F P 56 in this region? If so, describe a test that achieves this value. If not, explain. HW 5.2 In the binary communication system shown in Fig. 5.2, messages X 0 and X 1 occur with a priori probabilities 1/4 and 3/4, respectively. Suppose that we observe YXW  where   , W 3434 is statistically independent of whether message X 0 or X 1 occurs.
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RP-HW5 - Kyung Hee University Department of Electronics and...

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