Consider this joint distribution for random variables \(X\) and \(Y\), written in hierarchical form:

\[ \begin{aligned} X&\sim\text{Gamma}\left(\alpha,\,\beta\right)\\ Y\mid X=x&\sim \text{Exponential}(x). \end{aligned} \]

1. Sketch the joint range of \((X,\, Y)\).
2. What is the joint density of the random pair \((X,\, Y)\)?
3. What is the marginal density of \(Y\)?
4. What is the density of \(X\) conditional on the event \(Y=y\)? Is it recognizable?