# PS07 - University of Illinois Fall 2009 ECE 313: Problem...

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University of Illinois Fall 2009 ECE 313: Problem Set 7 Decision-Making Under Uncertainty Due: Wednesday October 14 at 4 p.m. Reading: Ross, Chapter 3; Powerpoint Lecture Slides, Sets 15-18 Decision-Making and Decision Making Under Uncertainty : Additional reading material available on the COMPASS web page for ECE 313 Noncredit Exercises: Chapter 3: Problems 53, 58, 59, 62, 63, 70-74, 78, 81 Theoretical Exercises 6, 7(a), 25, 26; Self-Test Problems 15-26. Reminder: Hour Exam I on Monday October 12, 7:00 p.m. – 8:00 p.m., 100 Noyes Lab. Coverage of material is Problem Sets 1-6 1. [A warm-up exercise] If H 0 is the true hypothesis, the random variable X takes on values 0, 1, 2, and 3 with probabilities 0.1, 0.2, 0.3, and 0.4 respectively. If H 1 is the true hypothesis, the random variable X takes on values 0, 1, 2, and 3 with probabilities 0.4, 0.3, 0.2, and 0.1 respectively. (a) Find the likelihood matrix L and indicate the maximum-likelihood decision rule by shading the appropriate entries in L . What is the false-alarm probability P FA and what is the missed-detection probability P MD for the maximum-likelihood decision rule? (b) Now suppose that the hypotheses have a priori probabilities π 0 = 0 . 7 and π 1 = 0 . 3. Use the law of total probability to ﬁnd the average error probability of the maximum-likelihood decision rule that you found in part (a). (c) Use the a priori probabilities given in part (b) to ﬁnd the joint probability matrix J and indicate on it the Bayesian decision rule, which is also known as the minimum-error-probability (MEP) or maximum a posteriori probability (MAP) decision rule. What is the average error probability of the Bayesian decision rule? Is it smaller or larger than the average error probability of the maximum-likelihood decision rule? In the latter case, provide a brief explanation as to why the

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## This note was uploaded on 11/08/2010 for the course ECE ECE 313 taught by Professor S during the Spring '10 term at University of Illinois at Urbana–Champaign.

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PS07 - University of Illinois Fall 2009 ECE 313: Problem...

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