Consider the hypotheses H_0:y_1=-d+w_1 H_1:y_1=d+w_1 where the scalar y_1is obse
ID: 3127567 • Letter: C
Question
Consider the hypotheses H_0:y_1=-d+w_1 H_1:y_1=d+w_1 where the scalar y_1is observed, where d not equal sign 0 is some known constant, and where the random variable w_1 represents measurement noise. Suppose that d = 1/2 and that the noise is discrete with distribution pw_1(w_1)={1/8 w_1=_1 3/8 w_1=0 1/2 w_1=1 Draw a fully-labelled sketch of the operating characteristic of the likelihood ratio test (LRT) for this hypothesis test. In the remainder of the problem, in addition to y_1 there is a second observation y_2 that behaves under the two hypotheses according to H_0: y_2= -d + W_2 H_1: y_2 = d + w_2, where d is the same as in (1), and where w_2 represents the measurement noise in y2. For general (arbitrary) p_Wl (-w_2) and d, show that when P_w2(-w_2) =p_w1(-w2), for all W_2, then (Pf, Pd) = (1 - beta, 1 - alpha) is on the operating characteristic of the LRT for y_2 if (P_F, P_D) = (alpha, beta) is on the operating characteristic of the LRT for y_1. Suppose that d and p_w1(.) are as given in part (a), and that (2) holds, and further suppose that we can choose (potentially randomly) between measuring either y_1 or y_2, but not both. Plot the efficient frontier of (Pf, Pd) operating points for such a scenario. (practice) Does there exist a conditional distribution p_w2/w_1(w_2/w_1)such that holds and perfect detection (P_F = 0, P_d = 1) is achievable when y_1 and y_2 are simultaneously observed? If so, specify such a conditional distribution; otherwise, explain why not.Explanation / Answer
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