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Consider the random variables:

$X$ = the first year rate of return(%) expected from project A, and

$Y$ = the first year rate of return(%) expected from project B.

The table below gives the joint probability function $P(x, y)$ :

$\begin{array}{lc} & \mbox{X} \\ \mbox{Y} & \begin{array}{c|cccc} & -10 & 0 & 20 & 30 \\ \hline 10 & & & & \\ 20 & & & & \\ \end{array} \end{array}$

Answer the following questions, rounding your answers to two decimal places where appropriate.

(a) Find the marginal probability function of Y . In particular, find
i. $P_{Y}(10) = P(Y=10)$
ii. $P_{Y}(20) = P(Y=20)$

(b) Find the probability that the return from project B is 20 percent given that the return from project A is 20 percent.

(c) What is the expected rate of return from project A, $E(X)$ ?

(d) What is the expected rate of return from project B, $E(Y)$ ?

(e) What is the covariance between $X$ and $Y$ ?

You can earn partial credit on this problem.