1. Shirley Johnson, portfolio manager, has asked you to analyse a newly acquired portfolio to determine its mean value and variability. The portfolio consists of 50 shares of Xylophone Music and 40 shares of Yankee Workshop. Analysis of pard history indicates that the share price of Xylophone Music has a mean of 25 and a variance of 121. A similar analysis indicates that Yankee has a mean share price of 40 with a variance of 225. Let X be the share price of Xylophone Music and Y the share price of Yankee Workshop. The value of the newly acquired portfolio is demoled by Z. (a) Express the value of the newly acquired portfolio Z in terms of X and Y. (b) What is the expected value of the portfolio? (c) We recall that for read numbers a, 6 and random variables X and Y , we have Var(aX + by) = a Var(X) + b' Var(Y) + ZabCo (X, F). and that when X and Y are independent their covariance is 0). (i) Assuming that the share prices X and Y' are independent, what is the variance of the portfolio? (ii) We assume now that the share prices are not independent, with a positive covariance given by Cov( X, Y) = 27.5. Compute the variance of the portfolio in that context. (iii) Suppose that the covariance between share prices is actually -27.5. Now what is the variance of the portfolio? (iv) Which situation of (i), (ii) and (ii) is mued desirable for the portfolio manager and why? 2. Over the past decade, the mean number of hacking attacks experienced by members of the Information Systems Security Association is / = 510 per year with a standard deviation of a = 14.28 attacks. The number of ablacks per year is normally distributed. We assume that the numbers of allacks Xi over the years i = 1, 2 ..- are independent. Support: nothing in this environment changes. (a) What is the distribution of the average number of attacks over the next 10 years X10 = !)" (b) Calculate the probability that the average number of ablacks over the next 10 years is more than 600 attacks? You may need the fact that +(19.93) 81. (c) Cite a rule from the lecture notes on Special Random Variables that could have been used to predict that the value of the above probability is very small. Explain. 3. An administrator for a large group of hospitals believes that of all patients 30% will generate bills that become at least 2 months overdue. A random sample of 200 patients is taken. (a) What is the exact didribution of the sample proportion? (b) Use the central limit theorem in order to find an approximation of the probability that the sample proportion is more than 0.33? 4. An ecomounit uses the price of a gallon of milk as a nuscore of inflation. She finds that the population mon price: is $352 per gallon and the population standard deviation is $0213. You decide to sample 40 convenience stores, collect their prices for a gallon of milk, and compute the mean price for the sample. (a) What is the probability that the sample mean is between $3.78 and $3.86? (b) What is the likelihood the sample mean is greater than $3.92? (c) What is the probability that the difference between the sample mean and the population nuwan is less than $0 01
