SET 3 ERCAN KARADAS (To be handed in at the beginning of the class, July 23. Randomly chosen two problem will be graded) (1) A contractor estimates the probabilities for the number of days required to complete a certain type of construction project as follows: (a) What is the probability that a randomly chosen project takes less than 3 days to complete? (b) Find the expected time to complete the project. (c) Find the standard deviation of time required to complete the project. (d) Suppose the contractor's project cost is made of two parts: a xed cost of $20,000 and $2,000 for each day taken to complete the project. Find the mean and standard deviation of total project cost. (e) If three projects are undertaken, what is the probability that at least two of them will take at least 4 days to complete? Assume that project completion times are independent. (2) A basketball team's star 3-point shooter takes six 3-point shots in a game. Historically, he makes 40% of all 3-point shots taken in a game. (a) (b) (c) (d) Which probability model ts the best into this case? Explain. Find the probability that at least two shots will be made. Find the probability that exactly three shots will be made. Find the mean and the standard deviation of the number of shots made. (e) Find the mean and the standard deviation of the total number of points scored as a result of these shots. (3) Suppose you are asked to analyze a stock portfolio that contains 5 shares of stock A and 10 shares of stock B. The joint probability distribution of the stock prices is shown in the table below (numbers at the end of each row and column denote marginal probabilities): 1 2 ERCAN KARADAS (a) Complete the missing joint and marginal probabilities in the table above. (b) Compute the expected value of this portfolio. (c) Compute the standard deviation of this portfolio. (4) FedEx delivers approximately one million packages a day between East Asia and the United States. A random sample of the daily number of package delivery failures over the past six months provided the following results: 15, 10, 8, 16, 12, 11, 9, 8, 12, 9, 10, 8, 7, 16, 14, 12, 10, 9, 8, 11. There was nothing unusual about the operations during these days in the sample and thus, the results can be considered typical. Using this data and your understanding of the delivery process answer the following: (a) What probability model should be used and why? (b) What is the probability of 10 or more failed deliveries on a typical future day? (c) What is the probability of less than six failed deliveries? (d) Find the number of failures such that the probability of exceeding this number is 10% or less. (5) A company services home air conditioners. It is known that times for service calls follow a normal distribution with a mean of 60 minutes and a standard deviation of 10 minutes. (a) What is the probability that a single service call takes more than 65 minutes? (b) Find the probability that a single service call takes between 50 and 70 minutes? (c) The probability is 0.25 that a single service call takes more than how many minutes? (d) Find the shortest range of times that includes 50% of all service calls. (e) A random sample of four service calls is taken. What is the probability that exactly two of them take more than 65 minutes? (6) Swapna David is a customer assistant consultant for ABC Information Systems, who provides assistance for computer users. The mean number of PROBLEM SET 3 3 calls per hour is 40 from across the United States and calls are independent. She has just answered a call and is scheduled to take the next call. What is the probability that she will have at least 3 minutes to obtain her cup of tea? (7) Financial Managers Inc. buys and sells a large number of stocks routinely for the various accounts that it manages. Portfolio manager Andrea Colson has asked for your assistance in the analysis of the Johnson Fund. A portion of this portfolio consists of 10 shares of stock A and 8 shares of stock B. The price of A has a mean of 10 and a variance of 16, while the price of B has a mean of 12 and a variance of 9. The correlation between prices is 0.3. (a) What are the mean and the variance of the portfolio value? (b) Andrea has been asked to reduce the variance (risk) of the portfolio. She oers to trade 10 shares of stock A and receives two oers from which she can select one: 10 shares of stock 1 with a mean price of 10, a variance of 25, and a correlation with the price of stock B equal to -0.2; 10 shares of stock 2 with a mean price of 10, a variance of 9, and a correlation with the price of stock B equal to 0.5. Which oer should she select