$($b$)$ Consider the joint density

$f(x,y)=2,0x1,0yx$

$(4$pts$)$ Write down the two conditional densities $f(x|y)$ and $f(y|x),$ carefully

specify the range of each conditional distribution.

$(8$pts$)$ Write down the detailed steps of simulating $(x,Y)$ by using Gibbs

sampling. Drop the first $30$ simulated $(x,Y)$ from the burn$-$in period, make a

scatter plot for another $200$ simulated $(x,Y)$ on the $x-y$ plane.

$[$Note: The domain of $(x,Y)$ is a triangular region.$]$