## 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?