Recall that if X and Y are independent random variables, then Cov(X, Y) =0. It follows that Corr(X, Y) = 0, that is, they are uncorrelated. This problem shows that the converse is not true in general. Let X and Y be independent random variables defined by with probability 1/2, Y with probability 1/2, with probability 1/2, with probability 1/2. Let Z := XY. Show that X and Z are uncorrelated, but not independent. (Hint: Consider the events {Z = 1} and {X = 0}.)