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final_practice_2009

# final_practice_2009 - University of Illinois at Chicago...

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University of Illinois at Chicago Department of Electrical and Computer Engineering EC 531: Detection and Estimation Theory Spring 2009 Final exam: PRACTICE NAME: This exam has 6 questions. You will be given 2 hours. You may use the 2 course textbooks but no other aides/notes. No calculators are permitted. No talking, passing notes, copying (and all other forms of cheating) is permitted. Make sure you explain your answers in a way that illustrates your understanding of the problem. Ideas are important, not just the calculation. Partial marks will be given. If something has been proven in class or in the book feel free to cite and use the result without a re-derivation. 1

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1. Consider the following pair of hypotheses concerning a sequence Y 1 , Y 2 , · · · , Y n of independent random variables H 0 : Y k ∼ N ( μ , σ 2 0 ) , k = 1 , 2 , · · · , N H 1 : Y k ∼ N ( μ 1 , σ 2 a ) , k = 1 , 2 , · · · , N where μ 0 , μ 1 , σ 2 0 , σ 2 1 are known constants. (a) Determine the likelihood ratio as a function of μ 0 , μ 1 , σ 2 0 , σ 2 1 and the quantities N k =1 Y 2 k and N k =1 Y k . (b) Describe the Neyman-Pearson test for the two cases μ 0 = μ 1 , σ 2 0 < σ 2 1 μ 0 < μ 1 , σ 2 0 = σ 2 1 (c) Find the threshold and the Receiver Operating Characteristics for the case μ 0 = μ 1 , σ 2 1 > σ 2 0 with N = 1. 2
2. Consider hypothesis testing problem based on the observation of y . (a) Assume that the two hypotheses H 0 and H 1 are equally likely, P 0 = P 1 = 1 / 2. The conditional distribution is given by p ( y |H 0 ) = 1 , 0 y 1 0 else p ( y |H 1 ) = 2 y, 0 y 1 0 else which may be shown graphically as Problem 3 (40 points)

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final_practice_2009 - University of Illinois at Chicago...

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