Question: 1. (35 marks) (Estimation vs. Detection) Let 1 with probability, -1 with probability, X and the noise Z Unif (-2,2) be independent random variables.

1. (35 marks) (Estimation vs. Detection) Let 1 with probability, -1 with probability, X and the noise Z Unif (-2,2) be independent random variables. Their sum Y = X + Z is observed. (a) Find the conditional pmf pxy (ay). Find the MMSE of X given Y and its MSE. (b) Suppose we use a decoder to decide whether X = 1 or -1. Using the pmf px|y(x|y) found in part (a), find the MAP decoder and its probability of error. Compare the MAP decoder's MSE to the minimum MSE.

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