Consider the trivariate random vector $(Y_1,Y_2,Y_3)$ with the density

$f(y_1,y_2,y_3)=c$exp$\{-(y_1+y_2+y_3+y_1{y}_2+2y_2{y}_3+4y_1{y}_3)\}$;$,y_{1}0,y_{2}0,y_{3}0.$

where $c>0$ is the normalizing constant so that the density integrates to $1.$

Problem $3($a$)[3$ points$]$

Obtain the distribution of conditional densities of $Y_i|Y_j,Y_k$ for all possible distinct values of $ijk.$ Show

your work.

Problem $3($b$)[5$ points$]$

Implement the Gibbs sampling algorithm to generate values from the joint density of $(Y_1,Y_2,Y_3).$ Specifically,

run a chain with $10,000$ points, and burn the first $1000$ initially generated values. Then, print the approximation

for the mean and covariance matrix of $(Y_1,Y_2,Y_3)$ based on your generated sample. Also, plot the histogram

of the marginals distribution of $Y_1$ only superimposed by a smoothed density curve. Set breaks $=50$ for the

histogram, and use the density$()$ function for the density curve to be superimposed.

Problem $3($c$)[$ points$]$

Apply the Metropolis$-$Hastings algorithm to generate values from $(Y_1,Y_2,Y_3)$ using a chain with $50,000$ points.

Set the proposal density to the trivariate normal density with covariance matrix equal to two times the

identity matrix $($i$.$e$.,2**I).$ Your code should implement the following:

Use the mvrnorm$()$ function in the package MASS to generate multivariate normal values.

Begin your simulation with seed $2024.$

Start with the initial value $y_1=y_2=y_3=1.$

Your code should compute the acceptance rate and print it at the end.

Burn the first $5000$ points from the chain, and plot the histogram of the marginals of $Y_1,Y_2,$ and $Y_3,$

superimposed by a smoothed density curve. Set breaks $=50$ for the histogram, and use the density$()$

function for the density curve to

be superimposed.

$!$ need rcode.