Question: Let O be a Bernoulli random variable that indicates which one of two hypotheses is true, and let P(O = 1) = p. Under

Let Θ be a Bernoulli random variable that indicates which one of two hypotheses is true, and let P(Θ = 1) = p. Under the hypothesis Θ the random variable X has a normal distribution with mean 0, and variance l . Under the alternative hypothesis e = 1, X has a normal distribution with mean 2 and variance 1. Consider the MAP rule for deciding between the two hypotheses, given that X 1. Suppose for this part of the problem that p = 2/3. The MAP rule can choose in favor of the hypothesis Θ = l if and only if z 2 c1-Find the value of ci 2. For this part, assume again that p = 2 /3. Find the conditional probability of error for the MAP decision rule, given that the hypothesis 0is true. error 3. Find the overall (unconditional) probability of error associated with the MAP rule for p 1/2. You may want to consult to standard normal table.

Let O be a Bernoulli random variable that indicates which one of two hypotheses is true, and let P(O = 1) = p. Under the hypothesis e = 0, the random variable X has a normal distribution with mean 0, and variance 1. Under the alternative hypothesis e = 1, X has a normal distribution with mean 2 and variance 1. Consider the MAP rule for deciding between the two hypotheses, given that X = r. 1. Suppose for this part of the problem that p = 2/3, The MAP rule can choose in favor of the hypothesis e = l if and only if z > e1. Find the value of c. CI = 2. For this part, assume again that p = 2/3. Find the conditional probability of error for the MAP decision rule, glven that the hypothesis e = 0 is true. P(error|e = 0) = 3. Find the overall (unconditional) probability of error associated with the MAP rule for p= 1/2. You may want to consult to standard normal table.

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