Consider a binary hypothesis testing problem in which we observe a random variable X with...

Consider a binary hypothesis testing problem in which we observe a random variable X with the following conditional PDFs specified, and shown in Figure.

Suppose that the two hypotheses H0 and H1 have prior probabilities P(H0) = 0.4 and P(H1)= 0.6, respectively. We are interested in designing a decision rule for declaring ‘H0’ or ‘H1, and analyzing its performance.

(a) Find the minimum-probability-of-error decision rule, i.e., the decision rule that minimizes P(H0,‘H1’) + P(H1,‘H0’). Simplify your answer as much as possible.

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(b) Indicate, by shading the appropriate regions of the conditional probability density plots in the figure, how you would calculate:

(i) the (conditional) probability of false alarm, PFA ; and

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(ii) the (conditional) probability of miss, PM .