This problem concerns three variables, $X, Y$, and $W$. The pair of random variables $(X,Y)$ is described by a hierarchical model; $X\sim N(\mu, \tau^2)$ and $Y\mid X \sim N(x, \sigma^2)$. The third variable is defined in terms of the first two that is $W = Y - X$.
1) Find the joint distribution of $X$ and $Y$.
2) Find the joint distribution of $X$ and $W$.
3) Are $X$ and $W$ independent?
4) What is the distribution of $W$?
5) What distribution does $X + W$ follow? Justify your answer.
For 1, I have tried using the mgf's to try and find a way to simplify the distribution but am wondering if I should simply apply the regular way of solving a conditional hierarchical model. For 2, I am trying to use a change of variable and using a $z$ and $w$ but not having much luck. For 3, They are not independent because $X$ is part of $W$. For 4, $W$ is a normal distribution but I am unsure of cleaning up the parameters. For 5, I am still getting a normal distribution but do not believe this is correct. Any help is greatly appreciated.