Consider a random pair \((Q,\,Z)\) of continuous random variables whose joint distribution is given by this hierarchy:

\[ \begin{aligned} Q&\sim \text{Gamma}(a,\,b)\\ Z\mid Q=q&\sim\text{GF}(k, 1/q). \end{aligned} \]

So conditionally, \(Z\) has the distribution introduced in Problem 13 above, with \(1/q\) serving as the second parameter. \(k>0\) is just a constant throughout.

1. What is the joint density of \((Q,\,Z)\)?
2. What is the marginal distribution of \(Z\)?
3. What is the conditional distribution of \(Q\) given \(Z=z\)?