(f) Consider the space of all real-valued random variables with finite-variance. Show that this is a vector space, and call it S. (g) Note that for X, Y c S, we may view E[XY] as a function mapping S x S - R. Show that this is a valid inner product on this vector space. (h) For any pair of finite variance random variables (X, Y), the conditional expectation E[Y(X] is a function of X that is known to satisfy the following property: for all functions d E[(Y - E[YIX])$(X)] = 0. Using this definition, prove that the mean squared error E[(Y -(X ))"] of estimating Y from X is minimized by choosing o(X) = E[Y [X]. L.e., the conditional expectation minimizes the mean squared error of estimation. Hint: Think about how we proved the orthogonality principle without necessarily trying to formally define a subspace