A binary communication system has transmitted signal X, the Bernoulli (1/2) random variable. At the receiver, we observe Y = VX + W, where V is a "fading factor" and W is additive noise. Note that V and W are exponential (1) random variables and that X, V, and W are mutually independent. Given the observation Y, we must guess whether X = 0 or X = 1 was transmitted. Use a binary hypothesis test to determine the rule that minimizes the probability Pe of a decoding error. For the optimum decision rule, calculate Pe.