Suppose our receiver must observe the random variable and then make a decision as to what message was sent. Furthermore, suppose the receiver makes a three- level decision as follows:
Decide 0 was sent if Pr (M = 0|X = x) ≥ 0.9,
Decide 1 was sent if Pr (M = 1|X = x) ≥ 0.9,
Erase the symbol (decide not to decide) if both Pr (M = 0|X = x) < 0.9, and Pr (M = 1|X = x) < 0.9,.
Assuming the two messages are equally probable, Pr (M = 0) = Pr (M = 1) = 1/2, and that σ2 = 1, find
(a) The range of over which each of the three decisions should be made,
(b) The probability that the receiver erases a symbol,
(c) The probability that the receiver makes an error (i. e., decides a “0” was sent when a “ 1” was actually sent, or vice versa).