Refer to Exercise 23. Assume that if m = 0, the value s = −1.5 is sent, and if m = 1, the value s = 1.5 is sent. The value received is X, where X = s + E, and E ∼ N(0, 0.64). If X ≤ 0.5, then the receiver concludes that m = 0, and if X > 0.5, then the receiver concludes that m = 1.

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. If the true message is m = 1, what is the probability of an error, that is, what is the probability that the receiver concludes that m = 0?

c. A string consisting of 60 1s and 40 0s will be sent. A bit is chosen at random from this string. What is the probability that it will be received correctly?

d. Refer to part (c). A bit is chosen at random from the received string. Given that this bit is 1, what is the probability that the bit sent was 0?

e. Refer to part (c). A bit is chosen at random from the received string. Given that this bit is 0, what is the probability that the bit sent was 1?

Refer to Exercise 23.

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).