In our simplified model of the tennis serve, the only goal of the server is to position the ball in the serve box as close as possible to the service line. Let Y be the position where the ball lands with respect to the service line. If Y is positive the serve is good, if negative it's a fault. We define a utility function as follows: utility (g): 4(8) = if y 30 if where 'gi is the position where the ball landed (position and utility are in arbitrary units). (a) Assume that the continuous RV Y is uniformly distributed over an interval of length _, centered at position y = M. The parameter _ is fixed, but the server can control M, hence the position of the entire interval. Compute the expected utility E (u (+) as a function of the interval's center M. Sketch out the graph of the expected utility as a function of M over a range of relevant positive and negative values of M. What is the optimal value of M, achieving highest expected utility? (b) Consider now the case where obeys the following density: if f(x) k(y-1+1) k(-4+1+) f y SM. if M-1/xyM where is a constant. Determine the value of which makes fly) a legitimate density. Derive the expected utility as a function of M. Finding the optimal value of M, which achieves the highest expected utility, is not easy in this case. However, try to argue that, contrary to case (a), this optimal value of Mis less than L/2.