Here is the joint density for a random pair:

\[ f_{XY}(x,\,y) = \frac{24}{7} x(2-y),\quad 0<x<1\text{ and }0<y<1-x. \]

1. Sketch the joint range of \((X,\,Y)\).
2. Find the marginal density of \(Y\).
3. Find the marginal density of \(X\).
4. Find the conditional density of \(X\) given \(Y=y\).
5. Use this to compute \(P(X>1/2\,|\,Y=1/3)\).
6. What is \(E(X\,|\,Y=y)\)?