Let be a random variable which is distributed uniformly on the interval [0, 2], i.e. P(a b) = (b a)/(2). Let C = cos() and S = sin(). In this exercise we will compute E(C | S) and E(S | C). We will need to use the fact (from Fourier Series) that the set {sinn() | n = 0,1,2,...} is dense in L2S, so to check that a random variable X is perpendicular to L2S it suffice to check that X,sinn() = 0 for n = 1,2,3,....

(a) Find E(C | S).