$**$ Please Provide R Code below thank you$**$

The following table shows an estimate of the joint distribution of dual$-$carner couple educational

For example, $0\mathrm{.}5\%$ of sampled dual$-$earner couples contain a husband with high school grad$-$

uation only and a wife with more than $4$ years of college; $11\mathrm{.}6\%$ of sampled married couples

contain a husband and wife who both have some college, but no degree. Values close zero are

represented by $(**).$

$2\mathrm{.}1(2$ point$)$ Let $x$ be a Bernoulli random variable that indicates whether husbands in dual$-$

carner couples have completed at least four years of college $($ic$.x=1$ if the husband has

$4$ ycars of college and$/$or more than $4$ years of college; and $x=0$ if the husband has not.

attended or completed college$).$ Define $Y$ similarly for wives in dual$-$carner couples. Using

'Source: Winkler, A$.$ E$.(1998).$ Earnings of husbands and wives in dual$-$earner families. Monthly Lab. Rev.,

$121,42.$ link.

the table above, compute the estimated joint distribution of $(x,Y)($that is$,$ the probabilities

$Pr(x=x,Y=y)$ for each possible value of $x$ and $y?$

$2\mathrm{.}2(2$ point$)$ Use your answer to part $2\mathrm{.}1$ to estimate the covariance between husband and

wife four year college completion within dual$-$carner couples? $($Hint: we've already soen in

lectures for Bernoulli random variables that $E[x]=Pr(x=1).$ Similarly, if both $x$ and $Y$

are Bernoulli, then $E[xY]=Pr(x=1,Y=1).$

$2\mathrm{.}3(2$ point$)$ Let $W$ indicate the number of individuals in dual$-$carner couples that have

completed four years of college. That is$,W$ can be equal to zero, one or two. What is the

estimated distribution of $W?$ What is the estimated mean and standard deviation of $W?$