Prof. Valerie R. Bencivenga Economics 329 November 7, 2012 MIDTERM EXAM #2 Instructions: Answer the questions below in a blue exam book. Number your answers. Show your work to receive credit. The exam consists of 9 questions worth 340 points, and will last two hours. This is an open book, open notes exam. You may use a calculator; you may not use any device with wireless capability. Good luck! (60 points) 1. The bivariate probability distribution of the quantities demanded of two products is as follows: X = quantity of product #1 1 3 5 2 .2 .05 0 Y = quantity 3 .15 .2 .15 of product #2 4 0 .05 .2 a. Are the quantities demanded of these two products independent? Explain why or why not. b. What is the expected quantity of product #1 demanded? What is the variance of the quantity of product #1 demanded? c. What is the expected quantity of product #1 demanded given that the quantity of product #2 demanded is 4? What is the variance of the quantity of product #1 demanded given that the quantity of product #2 demanded is 4? DO NOT CALCULATE FINAL NUMERICAL ANSWERS FOR PART C. d. What is the covariance between the quantity of product #1 demanded and the quantity of product #2 demanded? e. The price per unit of product #1 is 40. The price per unit of product #2 is 70. State revenue as a function of the quantities demanded of the two goods. f. What is expected revenue? What is the variance of revenue? (40 points) 2. A firm funds three research projects for one year (Projects A, B, and C). Each project may succeed or fail. This investment strategy by the firm may be interpreted as an experiment. a. What is the sample space? b. The probability that each project will succeed is .6. The outcomes of the three projects are independent. What is the probability distribution over the sample space? c. Define the random variable X = number of projects that succeed. Give the probability distribution of X. What is the expected number of projects that succeed in one year? What is the variance? d. Each project costs $1 million to fund. If a project succeeds, the firm will get $5 million in net revenues from manufacturing and selling the product. Profit equals net revenues minus costs. What is expected profit? What is the variance of profit? (30 points) DO NOT CALCULATE FINAL NUMERICAL ANSWERS. 3. A home improvement store decides to put an information booth in the center of the store. Management believes that the average number of customers who will arrive at the booth in a five minute period is 4, and that this average number will be constant over the business day. a. What is the probability that, over any five minute interval, exactly 4 people will arrive at the information booth? b. What is the probability that more than one person will arrive at the booth? c. The person assigned to the information booth gets called away for fifteen minutes to handle an emergency, leaving the booth unattended. Assuming that no one gets discouraged by the line, what is the probability that more than ten people will be waiting when the attendant gets back? You must submit the exam questions as well as your blue exam book, in order to receive a score on the exam. Tuck this question paper inside your blue book.(20 points) DO NOT CALCULATE FINAL NUMERICAL ANSWERS. 4. A company needed to downsize a department with 30 people. 12 of them are forty years of age or older, and 18 of them are younger than forty. 10 people were laid off, and management said the layoffs were done randomly. a. If the layoffs really were done randomly, what is the probability that 8 or more out of the 10 people laid off would be forty years old or older? b. How would your answer change if 15 people were laid off? (30 points) 5. A real estate investor buys two properties. Annual net income from the first property (an apartment building) is $12,000 times the number of apartments that are rented out, minus $25,000 in property taxes and maintenance expenses. The number of apartments that are rented out is a random variable, X, with mean 10 and standard deviation 2. Annual net income from the second property (a parking lot) is half of revenue (the management company takes the other half), minus $13,000 in property taxes and maintenance. Revenue is a random variable, Y, with mean $100,000 and standard deviation $10,000. The correlation between the number of apartments rented out (X) and revenue from the parking lot (Y) is .4. a. What is the real estate investor’s expected annual net income from these two properties? b. What is the standard deviation of annual net income from these two properties? Note that you will need to compute the covariance from the correlation in order to compute the standard deviation. c. What are the units of the covariance between X and Y? (40 points) 6. An officer of a micro-lending company in a developing country has made eight loans of $100 each. Let Xi be a random variable representing the rate of return on loan i, i = 1, ..., 8. The rate of return on loan i is a random variable because some borrowers delay or skip payments. Xi is approximately normally distributed with mean .10, and standard deviation .02. a. What is the probability that loan i will have rate of return in excess of .11? b. What is the probability that the loan officer’s eight loans will have an average rate of return in excess of .11? c. Find an interval (a, b) such that the probability is .99 that the average rate of return will fall in this interval. (Give the interval that is symmetric around the expected value of the average rate of return.) d. How many loans would the entrepreneur have to make in order for the probability to be at least .99 that the average rate of return will be between .09 and .11? (40 points) 7. Imagine you are a small farmer in a poor Latin American country. You are trying to decide whether to continue to cultivate a traditional strain of wheat or switch to a new “high-yield” variety. If you plant the traditional strain, the yield of your wheat field is approximately normally distributed with mean 50 bushels and standard deviation 5 bushels. If you plant the high-yield variety, the yield of your wheat field is approximately normally distributed with mean 60 bushels and standard deviation 12.5 bushels. a. Your family