Question: Problem 3: Let X be an continuous-valued exponential r.v. with unknown parameter 1. Consider the binary hypothesis given by X=1 H : H: = 0.25.

Problem 3: Let X be an continuous-valued

Problem 3: Let X be an continuous-valued exponential r.v. with unknown parameter 1. Consider the binary hypothesis given by X=1 H : H: = 0.25. (a) You measure X once. Determine the MAP test on the value of X as a function of P[H0] = = P. (b) For P[H] = 0.5, determine the probability of error for your test in part (a). (c) Suppose now you measure X twenty-five times in independent experi- ments. You compute the sample average as 25 Xi. 25 i=1 Assuming the law of large numbers is reasonable, determine an approx- imate MAP test on the value of for the value P[H.] =p= 0.5 for the binary hypothesis being considered. Problem 3: Let X be an continuous-valued exponential r.v. with unknown parameter 1. Consider the binary hypothesis given by X=1 H : H: = 0.25. (a) You measure X once. Determine the MAP test on the value of X as a function of P[H0] = = P. (b) For P[H] = 0.5, determine the probability of error for your test in part (a). (c) Suppose now you measure X twenty-five times in independent experi- ments. You compute the sample average as 25 Xi. 25 i=1 Assuming the law of large numbers is reasonable, determine an approx- imate MAP test on the value of for the value P[H.] =p= 0.5 for the binary hypothesis being considered

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