Solution_2_2007

Solution_2_2007 - ECE 154B Winter 2007 Homework#2 Solutions...

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ECE 154B Winter 2007 Homework #2 Solutions 1. Consider the 2 hypotheses problem where the a priori probabilities for the two hypotheses are: π 0 = ¾ and π 1 = ¼ . Let the conditional probability densities of the observable Y under the two hypotheses be: f Y|0 (y) = K 1 y for 0 < y < 3, f Y|0 (y) = 0 elsewhere and f Y|1 (y) = K 2 for 0 < y < 3, f Y|1 (y) = 0 elsewhere. (You must first determine the constants K 1 and K 2. ). a. Find the decision rule that minimizes the probability of error. K 1 =2/9 (from Homework 1). K 2 = 1/3. 0 f Y 0 y = 1 f Y 1 y 3 4  2 9 y = 1 4  1 3 y = 1 2 Choose H 0 if y > 1/2. Choose H 1 if y < 1/2. b. Calculate the probability of error for this decision rule. P error = 0 P error H 0  1 P error H 1 = 3 4 0 1 2 2 9 y dy  1 4 1 2 3 1 3 dy = 1 48 5 24 = 11 48 c. Verify your calculation of the probability of error by simulation. Matlab: x=rand(1,10000)>(3/4); for n=1:10000 if x(n)==0 y(n)=3*sqrt(rand); else y(n)=3*rand; end end z=y<(1/2); sum(x~=z)/10000 2. Consider the 4 hypotheses problem with 4 equally likely hypotheses (H 0 , H 1 , H 2 , H 3 ) . Under each of the 4 hypotheses assume that the observable Y is a Gaussian
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random variable with means (-3, -1, +1, +3) respectively.
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