where bc is the \"round down\" operator (that is, for example, b2.5c = 2, b 2.5c = 3, b 3c = 3). Find the probability mass function of Y . (Hint: For Y to take value k, what values should X take?) c) (6 pts.) Compute Var(Y ). 3. (30 pts.) Suppose Jill and Bill are each asked to turn a fortune wheel of known contents once. However, the result of the rolls are not revealed to the contestants. The contestants are then asked to whether take their unknown portion home or equally split what they collect. It is reasonable to assume that the prizes obtained by Jill ,X, and Bill ,Y , are independent identically distributed random variables drawn from the same set of N distinct prizes of value greater than equal to zero, each with probability pi , i 2 {1, , N }. The mean of the prizes on a roll of the wheel is and variance 2 . a) (10 pts.) Suppose the presenter announces the total prize X + Y . Could this information change the probability distribution of the dierence between Jill's prize and Bill's prize X Y ? If yes, describe a fortune wheel where it would. If not, prove that it wouldn't for any fortune ! wheel with N distinct prizes of value greater than equal to zero, each with probability pi , i 2 1 {1, , N }. (Extra (0 pts.): Suppose there are two random variables A and B. Prove that if knowing B changes the distribution of A conditional on B, (A|B), A and B cannot be independent.) b) (10 pts.) Dene the random variable U as U = aX +b, where a > 0 and b are deterministic p scalars. Compute the correlation coe cient Cor(U, X) = U,X = p Cov(U,X) . How does Var(U ) Var(X) your answer change if a < 0? c) (10 pts.) Compute the correlation coe cient between the total prize and the dierence between Jill's prize and Bill's prize in terms of and 2 . 4. (30 pts.) A company selling hot rolled steel operates a national warehouse in Tennessee to serve 10 customer locations dispersed around the country. The monthly demand at each customer location is normally distributed with mean 5 million metric tons and standard deviation 1 million metric tons. The demands at dierent customer locations are independent of each other. The steel inventory is replenished every month, and replenishment during the month is not desired. The company's goal is to cover all customer demand in every month at least 95% of the time. a) (10 pts.) Compute the minimum amount of steel that should be stocked in the warehouse at the beginning of each month, such that the company will be able to accomplish this goal. (Hint: Recall that if A and B are independent events, then P{A and B} = P{A} P{B}. In particular if X and Y are independent random variables, then P{X x and Y y} = P{X x} P{Y y}. =NORMSINV(0.5) function in Excel returns z such that P{N (0, 1) z} = 0.5. =NORMSDIST(1.5) function in Excel returns the probability P{N (0, 1) 1.5}.) b) (10 pts.) Primarily aimed to minimize its transportation costs, the company now considers operating a warehouse next to each customer and servicing each customer only from the warehouse next to them. The company's goal is still to cover all customer demand in every month at least 95% of the time with minimum inventory. In other words, the company would still like to see no customers lacking steel in 95% of the months. How much extra inventory needs to be placed nationwide? The steel inventory at each warehouse is replenished every month, and replenishment during the month, as well as shipping steel rolls from other warehouses is not desired: recall the aim of minimizing transportation costs. c) (10 pts.) Suppose the company has n customers with the same monthly demand distribution and the company decides to serve each of the n customers from warehouses next to each of these customers with the objective to cover all customer demand in every month at least 1 fraction of the time, with 0.05. In other words, the company would still like to see no customers lacking steel in 100(1 )% of the months. The steel inventory at each warehouse ! \f\f