6. Binary signal in Gaussian noise: Consider a binary communication channel in which the input X is either or c with probabilities 1/3 and 2/3, respectively. Here, > 0 is a given constant. The channel output is given by Y=X+N where NV is a standard normal random variable. For parts (c) and (d), you should provide closed-form expressions for your answers in terms of the standard normal CDF &. (a) Find the conditional PDF fy | x(y|z) of Y given X = x. What kind of random variable is this? (b) Find the marginal PDF fy(y) of Y. Plot fy(y) over 10 0 and c was transmitted otherwise. What is the probability of error for this receiver? What happens to this probability as c gets larger? Intuitively, why does this make sense? (d) What is the optimal receiver that minimizes the probability of error? What is the corre- sponding probability of error