6. (a) [15 POINTS] John has a ticket for a lottery with 100 tickets and one winner, with a prize of $3600, and a utility function given by U (X ) = log X, where X is a non negative random variable. (i) Calculate the expected value of the lottery ticket. (ii ) Calculate John's expected utility E (U (X) ) and his certainty equivalence. (iii) Is John risk averse? Justify your answer. (b) [14 POINTS] Let X and Y be 2 random variables with E (X) = -3, V (X) = 4, E(Y) = -1, V (Y) = 1 and Cor (X, Y) = -0.25. (i) Compute COV (X, Y). (ii) Let Z = 1+ 2X +3Y. Compute E (Z) and V (Z)