# 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).

**View Solution:**## Answer to relevant Questions

In this problem, we extend the results of Exercise 3.36 to the case when there are more than two possible messages sent. Suppose now that the communication system sends one of four possible messages, M = 1, M = 2, M = 3 and, ...Mr. Hood is a good archer. He can regularly hit a target having a 3- ft. diameter and often hits the bull’s- eye, which is 0.5 ft. in diameter, from 50 ft. away. Suppose the miss is measured as the radial distance from the ...A random variable has a CDF given by Fx (x) = (1 –e –z) u (x). (a) Find Pr (X > 3). (b) Find Pr (X < 5| X > 3). (c) Find Pr (X > 6 | X > 3). (d) Find Pr (|X –5| < 4||X –6| > 2). A random variable X has a characteristic function, ϕX (ω). Write the characteristic function of Y= aX+ b in terms of ϕX (ω) and the constants a and b. Derive a formula expressing the variance of a random variable in terms of its factorial moments.Post your question