Your next task is to analyze a proposed portfolio of Chevron, American Airlines, and ExxonMobil stocks. The forecast analyst in your team has provided you with a table that summarizes the distribution of returns for each firm $($Hint: this is an extended version of the "boom," "normal," and "recession" problem of slide $3$ in Modern Portfolio Theory$)(15$ points$)$

$\backslash$table$[[\backslash$table$[[$State of the$],[$Economy$]],$Probability,Chevron,$\backslash$table$[[$American$],[$Airlines$]],$ExxonMobil$],[$State $1,5\%,-8\mathrm{.}3\%,19\mathrm{.}4\%,-15\mathrm{.}0\%],[$State $2,10\%,-3\mathrm{.}9\%,8\mathrm{.}2\%,-10\mathrm{.}0\%],[$State $3,15\%,-1\mathrm{.}3\%,5\mathrm{.}0\%,-1\mathrm{.}8\%],[$State $4,20\%,0\mathrm{.}9\%,3\mathrm{.}2\%,0\mathrm{.}8\%],[$State $5,20\%,3\mathrm{.}0\%,1\mathrm{.}0\%,2\mathrm{.}8\%],[$State $6,15\%,5\mathrm{.}2\%,-1\mathrm{.}1\%,5\mathrm{.}4\%],[$State $7,10\%,8\mathrm{.}3\%,-3\mathrm{.}6\%,20\mathrm{.}0\%],[$State $8,5\%,14\mathrm{.}2\%,-5\mathrm{.}0\%,30\mathrm{.}0\%]]$

a$)$ Obtain the expected return, variance, and standard deviation of returns for each stock.

b$)$ Obtain the expected return, variance, and standard deviation of returns for a portfolio that invests $$300,000$ in Chevron, $$500,000$ in American Airlines, and $$600,000$ in ExxonMobil.