An investor has $$100,000$ available for $1-$year investment. The investor is weighing two options: a money market fund that gives a fixed annual return of $12\%$ and an investment plan with an annual rate of return that can be regarded as a random variable with values that depend on prevailing economic conditions. Based on the second plan's past history under a variety of economic conditions, a very reliable analyst has subjectively determined the following probabilities associated with several possible rates of return:

$\backslash$table$[[$Rate of Return,Probability$],[0\mathrm{.}3,0\mathrm{.}20],[0\mathrm{.}25,0\mathrm{.}20],[0\mathrm{.}20,0\mathrm{.}30],[0\mathrm{.}15,0\mathrm{.}10],[0\mathrm{.}10,0\mathrm{.}10],[0\mathrm{.}05,0\mathrm{.}10]]$

a$)$ Can the above table be regarded as a table which provides a probability distribution of the rate of return? If yes, why?

b$)$ Which investment plan should be selected, if the choice is to be made on the expected rate of return? Show work.

c$)$ If you chose option $2,$ what is the probability that the rate of return will be less than that of the first option? Show work.

d$)$ Suppose there were a third investment plan with return rates and associated probabilities as follows:

$\backslash$table$[[$Rate of Return,Probability$],[0\mathrm{.}23,0\mathrm{.}40],[0\mathrm{.}20,0\mathrm{.}40],[0\mathrm{.}18,0\mathrm{.}10],[0\mathrm{.}10,0\mathrm{.}10]]$

Between the second and the third investment plans, which should be selected? Explain carefully and show work.