Question

# Let Θ be a Bernoulli random variable that indicates which one of two hypotheses is true,...

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.
Suppose for this part of the problem that p=2/3. The MAP rule can choose in favor of the hypothesis Θ=1 if and only if xc1. Find the value of 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.

a) Observe that if is the rejection region of , then

Therefore

b) The probability of error given that is

c) Now the map rule for p = 1/2 would be (from the previous calculation)

Therefore the unconditional probability of error is