Question

A binary message m, where m is equal either to 0 or to 1, is sent...

A binary message m, where m is equal either to 0 or to 1, is sent over an information channel. 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, 0.66). 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? Round the answer to four decimal places.

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?

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