Consider a binary hypothesis in which there is a cost associated with each type of decision. In addition to the cost C'10 for a false alarm and C'01 for a miss, we also have the costs C'00 for correctly guessing hypothesis H0 and the C'11 for correctly guessing hypothesis H1. Based on the observation of a continuous random vector X, design the hypothesis test that minimizes the total expected cost
Show that the decision rule that minimizes minimum cost test is the same as the minimum cost test in Theorem 8.3 with the costs C'01 and C'10 replaced by the differential costs C'01 - C'11 and C'10 - C'00.
Theorem 8.3
Minimum Cost Binary Hypothesis
Test For an experiment that produces a random vector X, the minimum cost hypothesis test is
