Suppose that we have a pair of random variables (X, Y ) with a mixed discrete and continuous distribution as follows. Y is a binary {0, 1} random variable described by a pmf pY (1) = 0.5. Conditioned on Y = y, X is continuous with a Gaussian distribution with mean σ² and mean y, that is,

This can be thought of as the result of communicating a binary symbol (a "bit") over a noisy channel, which adds 0 mean variance σ² Gaussian noise to the bit. In other words, X = Y + W, where W is a Gaussian random variable, independent of Y. What is the optimum (minimum error probability) decision for Y given the observation X? Write an expression for the resulting error probability.