Consider a random pair \((X,\,Y)\) with a joint distribution given by this hierarchy:
\[ \begin{aligned} X &\sim \textrm{Beta}(a,\, b) \\ Y\mid X = x &\sim \textrm{Gamma}\left(a+b,\, \frac{c}{x}\right). \end{aligned} \]
- What is the joint range?
- What is the marginal density of \(Y\)?
- What is the conditional density of \(X\)?
New distribution!
If \(X\sim \textrm{Beta}(a,\, b)\), then \(\text{Range}(X)=(0,\,1)\) and the density is
\[ f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{a-1}(1-x)^{b-1},\quad0<x<1. \]
That’s new to you, so see if you can work with it.