Find the average error probability in Example 9.10

Example 9.10

Suppose that t he random variable X is transmitted over a communication channel. Assume that the received signal is given by Y = X +W, where W ∼ N(0,σ²) is independent of X. Suppose that X = 1 with probability p, and X = −1 with probability 1 −p. The goal is to decide between X = 1 and X = −1 by observing the random variable Y . Find the MAP test for this problem.