## Distributions with Hierarchical models

This problem concerns three variables, X, Y, and W. The pair of random variables (X,Y) is
described by a hierarchical model; X~$N(\mu, \tau^2)$ and Y|X ~ $N(x, \sigma^2)$. 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.