Suppose that X and Y are two random variables on the probability space (S,P), with E[X] = x, and E[Y] =y. Define the covariance of X and Y as: == - Cov(X,Y) =E[(X x) (Y Hy)] Note, we have Cov(X, X) = Var(X), and Cov(X,Y) = Cov(Y, X) and Cov(aX + bX2,Y) = a Cov(X1,Y)+b Cov(X2,Y). = (a) Show that if X and Y are independent random variables, (meaning P(X = a, Y = b) = P(X = a)P(Y = b) for all a, b R, then Cov(X,Y) = 0 (b) Suppose you flip a coin, and if it lands tails you flip another coin, and on heads do nothing. Let X be the random variable that is 1 if the first coin is heads, and 0 if the first coin is tails. Let Y be the random variable be 0 if the first coin is heads, and +1 if the second coin is heads, and -1 if the second coin is tails. Show that Cov(X, Y) = 0, but X and Y are not independent random variables. A summary of the outcomes and the random variables are presented in a table below: