(Solution) - A single observation X is to be taken from a -(2025 Original AI-Free Solution)

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A single observation X is to be taken from a continuous distribution for which the p.d.f. is either f0 or f1, where

and
On the basis of the observation X, it must be decided whether f0 or f1 is the correct p.d.f. Suppose that the prior probability that f0 is correct is 2/3 and the prior probability that f1 is correct is 1/3. Suppose also that the loss from choosing the correct decision is 0, the loss from deciding that f1 is correct when in fact f0 is correct is 1 unit, and the loss from deciding that f0 is correct when in fact f1 is correct is 4 units. If the expected loss is to be minimized, for what values of X should it be decided that f0 is correct?