Problem 1: You are the mechanical engineer in charge of maintaining the machines in a factory. The plant manager has asked you to evaluate a proposal to replace the current machines with new ones. The old and new machines perform substantially the same jobs, and so the question is whether the new machines are more reliable than the old. You know from past experience that the old machines break down roughly according to a Poisson distribution, with the expected number of breakdowns at 2.5 per month. When one breaks down, $1,500 is required to fix it. The new machines, however, have you a bit confused. According to the distributor's brochure, the new machines are supposed to break down at a rate of 3.0 per month (and do cost $1,700 to fix). (In either event, the number of breakdowns in any month appears to follow a Poisson distribution.) On the basis of this information, you judge that it is equally likely that the rate is 3.0 or 1.5 per month.
a. Based on the minimum expected repair costs, should the new machines be adopted?
b. Now you learn that a third plant in a nearby town has been using these machines. They have experienced 6 breakdowns in 3.0 months. Use this information to find the posterior probability that the breakdown rate is 1.5 per month.
c. Given the posterior probability, should your company adopt the new machines in order to minimize expected repair costs?
Problem 2: Your inheritance, which is in a blind trust, is invested entirely in McDonald's or in U.S. Steel. Because the trustee own several McDonald's franchises, you believe the probability that the investment is in McDonald's is 0.8. In any one year, the return from an investment in McDonald's approximately normally distributed with mean 14% and standard deviation 4%, while the investment in U.S. Steel is approximately normally distributed with mean 12% and standard deviation 3%. Assume that the two returns are independent.
a. What is the probability that the investment earns between 6% and 18% (i) if the trust is invested entirely in McDonald's, and (ii) the trust is invested entirely in U.S. Steel?
b. Without knowing how the trust is invested, what is the probability that the investment earns between 6% and 18%?
c. Suppose you learn that the investment earned more than 12%. Given this new information, find your posterior probability that the investment is in McDonald's.
Problem 3: An investor with assets of $10,000 has an opportunity to invest $5000 in a venture that is equally likely to pay either $15,000 or nothing. The investors utility function can be described by the log utility function U(x)=ln(x), where x is his total wealth.
a. What should the investor due?
b. Suppose the investor places a bet with a friend before making the investment decision. The bet is for 1000; if a fair coin lands heads up, the investor wins $1,000, but if it lands tails up, the investor pays $1,000 to his friend. Only after the bet has been resolved will the investor decide whether or not to invest in the venture. What is an appropriate strategy for the investor? If he wins the bet, should he invest? What if he loses the bet?
c. Describe a real life situation in which the individual might find it appropriate to gamble before deciding on a course of action.