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

\[ \begin{aligned} X&\sim\textrm{Gamma}\left(\frac{d_2}{2},\, \frac{d_2}{2}\right)\\ Y\,|\, X = x&\sim\textrm{Gamma}\left(\frac{d_1}{2},\, \frac{d_1}{2}x\right).\\ \end{aligned} \]

Do some serious “massage and squint” to show that the marginal pdf of \(Y\) is

\[ f_Y(y)=\frac{\Gamma\left(\frac{d_1}{2}+\frac{d_2}{2}\right)}{\Gamma\left(\frac{d_1}{2}\right)\Gamma\left(\frac{d_2}{2}\right)}\left(\frac{d_1}{d_2}\right)^{\frac{d_1}{2}}y^{\frac{d_1}{2}-1}\left(1+\frac{d_1}{d_2}y\right)^{-\frac{d_1+d_2}{2}},\quad y>0. \]

This means that \(Y\) has the F-distribution.