The input to a back box is a random signal X which satisfies P(X = 1) = 1 P(X = 1) = p, for a constant p (0, 1). The output of the black box is a random variable Y , which is equal to the input plus some independent random noise. This noise has a Gaussian distribution with mean 0 and variance 2 > 0.

(a) Describe the joint distribution of X and Y by writing down expressions for P(a Y b and X = i) where a b and i {1, 1}. Calculate the conditional probabilities P(X = i|Y = y) for i {1, 1} and y R.

(b) Suggest a rule for predicting the value of X if the value of Y that is output from the box is y. ( You can assume the values of p and 2 are known to you). Explain why your suggested rule is good one by considering the probability that it correctly predicts the value of X. [ Hint: use the fact that y P(X = 1|Y = y) is an increasing function.]