the question is:

In this problem, we study a simple noisy communication channel. Suppose that X is a binary signal that takes value -1 and 1 with equal probability. This signal X is sent through a noisy communication channel, and the medium of transmission adds an independent noise term. More precisely, the received signal is Y = X + N, where N is standard normal, indpendendent of X. The decoder receives the value y of Y, and decides whether X was 1 or -1, using the following decoding rule: it decides in favor of 1 if and only if P(X = 1|Y = y) > 2P (X = 0| Y = y) . It turns out that the decoding rule can be expressed in the form: decide in favor of 1 if and only if Y > t, for some threshold t. Find the threshhold t. As an intermediate step, find p1 = P(X = 1|Y = y). P1= Now find t. t =