Recently, you have been considering investing your money to increase your earnings. There are $3$ investment plans for you to consider. Each plan calls for an investment of $$25,000$ and the return will be one year later. Plan A will return $$27,500.$ Plan B will return $$27,000$ or $$28,000$ with probabilities $0\mathrm{.}4$ and $0\mathrm{.}6,$ respectively. Plan C will return $$24,000,$ $$27,000$ or $$33,000$ with probabilities $0\mathrm{.}2,0\mathrm{.}5,$ and $0\mathrm{.}3,$ respectively.

a$)$ If the random variables $A,B,C$ denote the returns from investment plans $A,B,$ and $C,$ respectively, what are the mean and variance of the three random variables?

b$)$ Which plan would you choose if your objective is to maximize your expected return, and you also do not want to lose money? Provide justification for your choice.

$(20$ points$)$