Letbe a Bernoulli random variable that indicates which one of two hypotheses is true, and letP(=1)=p. Under the hypothesis=0, the random variableXhas a normal distribution with mean0, and variance1. Under the alternative hypothesis=1,Xhas a normal distribution with mean2and variance1.

Consider the MAP rule for deciding between the two hypotheses, given thatX=x

.

1.Suppose for this part of the problem thatp=2/3. The MAP rule can choose in favor of the hypothesis=1if and only ifxc1. Find the value ofc1

c1=

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

P(error|=0)=

3.Find the overall (unconditional) probability of error associated with the MAP rule forp=1/2.