Consider a binary hypothesis testing problem in which a receiver observes a random variabl...

Consider a binary hypothesis testing problem in which a receiver observes a random variable R. Based on this observation the receiver decides which one of two hypotheses—denoted by H0 and H1—to declare as true. The receiver can be tuned to operate at any point on the receiver operating characteristic, which for this receiver is given by PD = where PD = P(‘H1’|H1) and PFA = P(‘H1’|H0). (As a reminder, the probability of error Pe of the receiver is defined as the probability of declaring ‘H0’ and having H1 true, or declaring ‘H1’ and having H0 true.)

(a) For this part, suppose that the prior probability of hypothesis H0 being true is P(H0) =  and that the receiver is tuned to operate at the point PD =  on the ROC curve. Determine PFA and the probability of error Pe at that operating point.

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(b) For the prior probability of H0 given in (a) (i.e., P(H0) = ), there is an operating point on the ROC curve that minimizes the overall probability of error Pe. Determine PD if the receiver operates at that point.

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(c) Now let P(H0) =  . Determine PD and PFA on the ROC curve and the corresponding Pe such that Pe is minimized