exercise 1_questions

Long Commute (Y=0) Short Commute (Y=1) Total
a) Compute (i) E(Y) and E(X); (ii) var(Y) and var(X); and cov(X, Y) and corr(X,Y). b) Consider two new random variables W= 3 + 6X and V=20 – 7Y. Compute (i) E(W) and E(V); var(W) and var(V); and cov(W,V) and corr(W,V).
1. The discrete Poisson distribution has one parameter λ. Its density function is given by
f ( y)Baidu Nhomakorabea
y e
y!
y 0,1,2,...
where y! 1x2 x3x...xy . a) Given that this density function is valid, what properties must it satisfy? b) Assuming 2 , calculate the following; i. Prob(y=2) ii. Prob(y<2) iii. Prob(y>2) c) Sketch out the shape of the density function if 2 . Is it skewed or symmetric? Explain. 2. The following table gives the joint probability distribution of the discrete random variables weather conditions (X) and the commuting time of a student commuter (Y). Let X be a binary random variable that equals 0 if it is raining and 1 if it is not. Similarly, Y is a binary random variable that equals 1 if the commute is short (less than 20 minutes) and equals 0 otherwise. Rain (X=0) 0.15 0.15 0.30 No Rain ((X=1) 0.07 0.63 0.70 Total 0.22 0.78 1
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3. The following table gives the joint probability distribution between employment status and college graduation among those either employed or looking for work (unemployed) in the working age U.S. population, based on the 1990s U.S. Census. Unemployed (Y=0) 0.045 0.005 0.050 Employed (Y=1) 0.709 0.241 0.950 Total 0.754 0.246 1.000
4. Assignment: Suppose you have some money to invest –for simplicity £1- and you are planning to put a fraction w into a stock market mutual fund and the rest 1-w, into a bond mutual fund. Suppose that £1 invested in a stock fund yields R s, after one year and that £1 invested in a bond fund yields Rb, that Rs is random with mean 0.08 (8%) and standard deviation 0.07, and that Rb is random with mean 0.05 (5%) and standard deviation 0.04. The correlation between Rs and Rb is 0.25. If you place a fraction w of your money in the stock fund and the rest 1-w, in the bond fund, then the return on your investment is R=wRs + (1-w)Rb. a) Suppose that w=0.5. Compute the mean and standard deviation of R. b) Suppose that w=0.75. Compute the mean and standard deviation of R. c) What value of w makes the mean of R as large as possible? What is the standard deviation of R for this value of w? d) What is the value of w that minimizes the standard deviation of R? (You can show this using a graph, algebra, or calculus).
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UNIVERSITY OF BRISTOL School of Economics, Finance and Management MSc in Economics, Finance and Management Quantitative Methods-Statistics
Exercise 1 Note: Bring your solutions to questions 1-3 to the tutorial in week 3. Solution to question 4 is part of the assignment to be handed in at the end of the term (week 10).
Non college grads (X=0) College grads (X=1) Total
a) Compute E(Y). b) The unemployment rate is the fraction of the labour force that is unemployed. Show that the unemployment rate is given by 1 – E(Y). c) Calculate E(Y/X=1) and E(Y/X=0). d) Calculate the unemployment rate for (i) college graduates and (ii) non-college graduates. e) A randomly selected member of this population report being unemployed. What is the probability that this worker is a college graduate? A non-college graduate? f) Are educational achievement and employment status independent? Explain.