Your name (please print it) ___________________________________Econ 120A Ramu RamanathanSpring 2003 First Midterm (20%)Your name (please print it) ___________________________________Your Student Id. (NOT Soc. Sec. no.)___________________________DO NOT TURN THIS PAGE UNTIL EVERYONE HAS RECEIVED THE EXAM AND YOU ARE GIVEN THE SIGNAL TO START. ALSO, YOU MUST STOP WRITING WHEN YOU ARE ASKED TO DO SO (YOU WILL BE GIVEN A 2 MINUTE WARNING). TEN POINTS WILL BE DEDUCTED FOR EACH MINUTE OF EXTRA TIME IT TAKES YOU TO STOP WRITING.If you use a pencil, you forfeit the right to complain about grading UNLESS YOU PICK UP THE EXAM FROM THE TA FROM HIS/HER OFFICE AND LOOK AT THE GRADING BEFORE LEAVING THE OFFICE. Make sure that all pages (1 through 5) are there. Read the questions carefully and make sure that you do not misunderstand them. If you get stuck somewhere, don't waste time but move on.I CONSIDER CHEATING AS A VERY SERIOUS MATTER AND WILL GIVE ANF IN THE COURSE TO ANY ONE CHEATING AND ALSO REFER HIM/HER TO THE DEAN FOR DISCIPLINARY ACTION. I HAVE DONE SO IN THE PAST.MAXIMUM NUMBER OF POINTS = 50THE NUMBER OF POINTS FOR EACH PART IS AN INDICATION OF THE MAXIMUM NUMBER OF MINUTES THAT PART WOULD REQUIRE TO ANSWER.1I. ( 5+3+7 =15 points)The continuous random variable X has the geometric distribution with f(x) = a xa-1 with a > 0 and 0 < x < 1.(a) Derive an expression for the mth moment E(Xm).(b) From that write down E(X) and E(X2), in terms of a.(c) Use these to derive the variance of the distribution.II. (6+2+6+2 = 16 points)In the TV show “Who wants to be a millionaire”, a contestant has earned $64,000 and is trying for $125,000. However, she does not know which of the four answers iscorrect. If she guesses correctly, she will get $125,000. If she is wrong, she ends up with $32,000. Her dilemma is whether to walk away with $64,000 (Option A) or takea gamble and choose one answer completely at random (Option B) and with the equal probability of ¼. (a) Compute the expected win for each option. Carefully explain.Option A (walk away with $64,000):2Option B (gamble for $125,000):(b) If the goal is to maximize the expected value of a win, what is your advice to her? Should she gamble or walk away? Why? Suppose, instead of the above scenario, she used the 50/50 lifeline and narrowed the answers to two and wants to guess at random with the equal probability of ½ amongthe remaining two answers (Option C). (c) Compute the expected win for this option. (d) Now what is your advice, based on maximizing the expected value of wins, between walking away with $64,000 and choosing Option C?3III. (4+3+3+4+5 = 19 points)The probability of a male, age 60, dying within one year is 0.025 and the probability of a female, age 55, dying within one year is 0.01 and the two events are independent. If a man and his wife are ages 60 and 55 respectively,(a) What is the probability that they both die?(b) What is the probability that they both survive?(c) What is the probability that only one will die within a year?(d) Suppose you sell a one year term life insurance of $25,000 to each, that is,you pay $25,000 to the surviving spouse if only one of them dies, $50,000 to their children if both of them die, and nothing if both live through the year. Calculate the expected total payment.(e) If you want to make an expected total profit of $200, what premium should you charge each of them assuming that the premiums are different and in proportion to the respective probabilities of