# Question: Suppose our receiver must observe the random variable and then

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).

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).

## Relevant Questions

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