Refer to image. Please write using R code. Problem $3.$ Suppose that $(x,Y)$ follows a bivariate normal distribution with a joint density

$f(x,y)=\frac{1}{2{}_1{}_2\sqrt[\phantom{2}]{1-{}^2}}$exp$[-\frac{1}{2(1-{}^2)}\{\frac{(x-{}_1{)}^2}{{}_1^2}-\frac{2(x-{}_1)(y-{}_2)}{{}_1{}_2}+\frac{(y-{}_2{)}^2}{{}_2^2}\}]$

where ${}_1=1,{}_2=2,{}_1=1,{}_2=2,$ and correlation $=0\mathrm{.}5.$

$($a$)$ Write $R$ code for a $3$D plot of $f(x,y)$ vs$.(x,y)$ over the grid points grid. $,$

$3,$ by$=0\mathrm{.}1)$ for $x$ and grid. $,$ by $=0\mathrm{.}1$ for $y.$

$($b$)$ To sample $(x,Y)$ from this joint distribution, one can use the following algorithm:

$($i$).$ generate $x$ from its marginal distribution $N({}_1,{}_1^2)$;

$($ii$).$ generate $Y$ given $x$ from a conditional density

$Y|x=xN({}_2+\frac{{}_2}{{}_1}(x-{}_1),(1-{}^2){}_2^2).$

Use this algorithm to generate $n=10,000$ pairs of random values from this joint

distribution.

$($c$)$ Based on the $10,000$ pairs of random values, calculate the sample correlation coefficient

$r$ between the $x-$ and $Y-$samples and also evaluate the following probabilities:

$($c

$($c