Texts: 6. (15 points) Suppose a transmitter sends S to be a random variable defined as
[1, w.p. 3; S = {0, w.p. 4; +1, w.p. 3].
The receiver has a noise observation
Y = S + Z
where Z is a Laplacian noise with a pdf of fz(z) = e^(-|z|) for -โ < z < โ. We assume that S and Z are statistically independent.
(a) (5 points) Find fys(y|s) for s = -1, 0, 1. Sketch the conditional pdfs on the same graph.
(b) (5 points) Find the optimal decoding rule for classifying the signal whether S is -1, 0, or +1. Give your answer in terms of ranges of values of Y.
(c) (5 points) Find the probability of decoding error in terms of P(S = -1), P(S = 0), and P(S = +1).
Note: The formatting and presentation of the text have been corrected.