An investor wishes to invest \(\$ 1,000\) and is examining two investment prospects. The net dollar 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

\(\mathrm{E}\left[\begin{array}{l}X_{1} \\ X_{2}\end{array}ight]=\left[\begin{array}{l}.15 \\ .07\end{array}ight]\) and \(\operatorname{Cov}(\mathbf{X})=\left[\begin{array}{cc}.04 & -.001 \\ -.001 & .0001\end{array}ight]\).

(a) If the investor invests \(\$ 500\) in each project, what is his/her expected net dollar return? What is the variance associated with the net dollar return?

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