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$)$:

YX$1020100\mathrm{.}110\mathrm{.}0800\mathrm{.}090\mathrm{.}05200\mathrm{.}070\mathrm{.}27300\mathrm{.}120\mathrm{.}21$XY$1002030100\mathrm{.}110\mathrm{.}090\mathrm{.}070\mathrm{.}12200\mathrm{.}080\mathrm{.}050\mathrm{.}270\mathrm{.}21$

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$.(10)=(=10)$PY$(10)=$P$($Y$=10)$

ii$.(20)=(=20)$PY$(20)=$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 $?$