uantitative Methods for MBAs Homework 4 Due September 13 in class This homework covers the basic concepts of probability, discrete and continuous random variables. Please answer using the space provided. Question 1 (20 points) An investor forms a portfolio using 2 stocks. He puts 25% of money into stock 1 and 75% of money on stock 2. The investor knows that the expected return of stock 1 is E(R1 ) = 0.08, while the expected return of stock 2 is E(R2 ) = 0.15. The standard deviations of each stock are respectively 1 = 0.12 and 2 = 0.22. Part A. (5 points) Compute the expected value of the portfolio 1 Part B. (5 points) If the two stocks are independent, what is the variance of the portfolio return? Part C. (5 points) If the two stocks have correlation 12 = .5, what is the variance of the portfolio return? Part D. (5 points) If the two stocks have correlation 12 = 1, what is the variance of the portfolio return? 2 Question 2 (30 points) Let Z denote the random variable distributed according to a standard normal distribution, what is the expected value, variance and standard deviation of the following variables? Explain your answers. 1. (5 points) X = 5Z 2. (5 points) X = 1 + 4Z 3. (5 points) X N (2.4, 2.25) 3 4. (5 points) X N (4.4, 9) 5. (5 points) Consider two independent variables, X1 = 3 + 0.4Z, and X2 = 5 + 0.9Z. Find the mean, standard deviation and variance of Y = 2X1 + 6X2 6. (5 points) Consider two independent variables, X1 = 2 + 2Z, and X2 = 7 + 5Z. Find the mean, standard deviation and variance of Y = 4X1 + 5X2 4 Question 3 (20 points) The joint probability distribution of X and Y is partially shown in the table below. 2 X 0 0.1 3 Y 1 2 0.4 0.1 5 0 0.2 0.2 In addition you know the conditional probability P (Y = 2|X = 2) = 0.25 P (X = 5|Y = 0) = 0.5 1. (8 points) Complete the table by computing all the joint probabilities and all the marginal probabilities. Show your computations step-by-step. 5 2. (2 points) Compute the expected value of X (show your math) 3. (2 points) Compute the variance of X (show your math) 4. (2 points) Compute the expected value of Y (show your math) 5. (2 points) Compute the Variance of Y (show your math) 6 6. (2 points) Compute the covariance of X and Y (show your math) 7. (2 points) Are X and Y independent? Explain how you get the result. 7 Question 4 (20 points) To answer this question you cannot use the computer. You need to come up with the answers using only the data provided. Here are several probabilities for the standard normal distribution. P (Z < 0.97) = 0.834 P (Z < 1.21) = 0.8869 P (Z > 1.13) = 0.8708 Using only these probabilities and the properties of the distribution, compute the following probabilities. Explain your answers and which properties of the distribution you used. 1. (5 points) P (Z < 1.13) 2. (5 points) P (Z > 1.21) 8 3. (2 points) P (Z > 0.97) 4. (2 points) P (1.13 < Z < 1.21) 5. (2 points) P (0.97 < Z < 1.21) 9 6. (2 points) For the variable X = 13+4Z, compute the probability P (16.88 < X < 17.52) 7. (2 points) For the variable X = 4 + 0.8Z, compute the probability P (3.224 < X < 3.096) 10 Question 5 (10 points) Facebook is deciding how to price an ad for a specific company. You are in charge of the analysis. You know that only 23% of all facebook users that see the ad of the company click on it, generating revenue for Facebook. If the company runs a campaign targeting a random sample of 10000 users (i.i.d.), and assuming that the revenue for each click is $0.35, what is the expected revenue of the campaign? What is the variance of the revenues? 11