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spring98exam2[1]

# Answer the likelihoods of the three events under the

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Unformatted text preview: ikelihoods of the three events under the 2 hypotheses are calculated basically as di erent areas under a Gaussian CDF: Under H0: 1 P (A2jH0) = P f;1 S0 < +1g = P f; 2 1 P (A1jH0) = P fS0 < ;1g = P f S0 < ; 2 g = 1 ; (1=2) = 0:308 2 1 g = 1 ; (1=2) = 0:308 2 Under H1: P (A3jH0) = P fS0 +1g = P f S0 2 S0 < 1 g = 2 (1=2) ; 1 = 0:384 2 2 ; P (A3jH1) = P fS1 +1g = P f S1 2 1 P (A2jH1) = P f;1 S1 < +1g = P f ;12; 1 ; P (A1jH1) = P fS1 < ;1g = P f S1 2 1 < ;12; 1 g = 1 ; (1) = 0:159 1 ; 1 g = 1 ; (0) = 0:5 2 S1 ; 1 < 1 ; 1 g = (0) ; (;1) = 0:341 2 2 A1 H0 H1 0:308 0:159 A2 0:384 0:341 A3 0:308 0:5 b. Find the maximum-likelihood decision rule for the likelihood matrix computed in part (a), and indicate it on the matrix below. Answer: The ML decision rule for each event is speci ed by boxing the entry for the appropriate hypothesis. A1 H0 H1 0.308 0:159 A2 0.384 0:341 A3 0:308 0.5 c. Under this rule, what is the probability of deciding H0 given that H1 is true? Answer: From the above table, P (1 ! 0) = 0:159 + 0:341. P (1 ! 0) = 0:5 d. Now suppose that the prior probabilities P (H0 ) = 0:2 and P (H1) = 0:8 are known. Find the Bayes decision rule for this case, and indicate it on the matrix below. Answer: If we now know the prior probabilities of H0 and H1, we can get the joint probability matrix from the ML matrix by multiplying each row by the corresponding prior. A1 H0 H1 0:0616 0.1272 A2 0:0768 0.2728 A3 0:0616 0.4 e. What is the average probability of error for the above Bayes decision rule? Answer: We see that Bayes' Rule has us deciding in favor of H1 for all 3 events. Therefore, th...
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