Question: Let
Y = X + V
where X, and V are independent random variables. X takes the values +1 and -1 with equal probability, and V is a zero-mean Gaussian random variable with variance σ2, respectively
(a) Evaluate the a-posteriori probabilities P[X = 1 | Y ] and P[X = -1 | Y ].
(b) Find the maximum a-posteriori estimate XˆMAP(Y ) of X given the observation Y .
(c) Find the Bayes least-squares estimate XˆMSE(Y ) of X given Y .