Question

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=

Find t.

t=

Answer 1