A binary message m, where m is equal either to 0 or to 1, is sent over an information channel. Because of noise in the channel, the message received is X, where X = m + E, and E is a random variable representing the channel noise. Assume that if X ≤ 0.5 then the receiver concludes that m = 0 and that if X > 0.5 then the receiver concludes that m = 1. Assume that E ∼ N(0, 0.25).
a. If the true message is m = 0, what is the probability of an error, that is, what is the probability that the receiver concludes that m = 1?
b. Let σ2 denote the variance of E. What must be the value of σ2 so that the probability of error when m = 0 is 0.01?