Let y x v where x and v are independent random variables

Question

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 .