shares. Denote next year’s value of one unit of investment in bonds as X and one unit of investment in shares as Y. Suppose that the two types of investment are independent. The marginal density functions for X and Y are, respectively, given in table 1 and table 2: Table 1 x 24 25 26 P(X = x) 0.2 0.3 0.5 Table 2 y 48 49 50 51 P(Y=y) 0.1 0.2 0.4 0.3 a) Determine the joint probability density function of X and Y. b) Suppose that you consider building a portfolio that mixes both bonds and shares. Also suppose that the portfolio’s return can be described by the following function:                          U =50X −30Y −15. What are the expected return and the variance of the portfolio?

is 20. Determine d so that the average value of the function y = 3 x + 2 40 35 30 25 20 15 10 2 4 6 10 12

a and b

2. Measurements were taken for entropy and enthalpy changes for many salts under a variety of water + methanol of NaCl (table salt) for several concentrations of methanol. Four independent conditions. The following table presents the results for enthalpy of transfer (in kJ/mol) from water to measurements were made at each concentration. Concentration (%) Enthalpy 1.62 1.60 1.62 1.66 5 10 20 30 2.69 2.66 2.72 2.73 3.56 3.45 3.65 3.52 3.35 3.18 3.40 3.06 a. Construct an ANOVA table in Excel. (8 points) b. Is it plausible that the enthalpy is the same at all concentrations? Explain. (4 points) No, brause the value is very low c. Use the Fisher LSD method to find a 95% confidence interval for the difference between the means for concentrations 5 and 30. Show your work. (12 points) d. Compute the quantity s = √MSE, the estimate of the standard deviation σ. (2 points)