An investor has \(\$ 10,000\) to invest between two investment projects. The rate of return per dollar invested in the two projects can be represented as the outcome of a bivariate random variable \(\left(X_{1}, X_{2}ight)\) where

\(\mathrm{E}\left[\begin{array}{l}X_{1} \\ X_{2}\end{array}ight]=\left[\begin{array}{l}.20 \\ .05\end{array}ight]\) and \(\operatorname{Cov}(\mathbf{X})=\left[\begin{array}{cc}.04 & .002 \\ .002 & .0001\end{array}ight]\)

(a) If the investor invests \(\$ 5,000\) in each project, what is the expected dollar return? What is the variance associated with the dollar return?

(b) If the investor wishes to maximize expected dollar return, how should the money be invested?

(c) If the invest wishes to minimize variance of dollar returns, how should the money be invested?

(d) Suppose the investor wishes to invest the \(\$ 10,000\) so that his/her expected utility is maximized, where \(\mathrm{E}(U(R))=\mathrm{E}(R)-.01 \operatorname{var}(R)\), where \(R=\alpha_{1} X_{1}+\alpha_{2} X_{2}\) represents the total return on the investment, \(\alpha_{1}+\alpha_{2}\) \(=10,000\), and \(\alpha_{i} \geq 0\) for \(i=1,2\). How much money should he/she invest in each of the two projects?