Similar to the last example, suppose that X = 0 +W, where O is equally likely to take the values 1 and 1, and where W is standard normal noise, independent of O. We use the estimator 0, with O = 1 if X > 0 and O = -1 otherwise. (This is actually the MAP estimator for this problem.) a) Let us assume that the true value of O is 1. In this case, our estimator makes an error if and only if W has a low (negative) value. The conditional probability of error given the true value of O is 1, that is, P(O +1|0 = 1), is equal to 4(-1) O (0) O (1) where & is the standard normal CDF. b) For this problem, the overall probability of error is easiest found using the formula O P( + 0) = S P(@ +0|X = x)fx(x) da O P( + 0) =E, P(@ + 0|e = 0) po (0)