A binary communication system has transmitted signal X, a Bernoulli (p = 1/2) random variable. At the receiver, we observe Y = V X + W, where V is a "fading factor" and W is additive noise. Note that X, V and W are mutually independent random variables. Moreover V and W are exponential random variables with PDFs
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 PERR of a decoding error. For the optimum decision rule, calculate PERR.