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=0, the random variable X has a normal distribution with mean 0, and variance 1. Under the alternative hypothesis Θ=1, X has a normal distribution with mean 2 and variance 1.

Question

Consider the MAP rule for deciding between the two hypotheses, given that X=x.

Suppose for this part of the problem that p=2/3. The MAP rule can choose in favour of the hypothesis Θ=1 if and only if x≥c1. Find the value of c1.

c1=

For this part, assume again that p=2/3. Find the conditional probability of error for the MAP decision rule, given that the hypothesis Θ=0 is true.

P(error|Θ=0)=

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.

Answer 1

Has anyone solved this question?

Answer 2

c1 = 2/3

Answer 3

1. 0.653
2. 0.257
3. 0.159

Answer 4

How did you get 0.653?