Chapter 5.2, Problem 2E

Let $X_1$ and $X_2$ be independent chi-square random variables with $r_1$ and $r_2$ degrees of freedom, respectively. Let $Y_1=\frac{\left(\frac{X_1}{r_1}\right)}{\left(\frac{X_2}{r_2}\right)}$ and $Y_2=X_2$.

(a) Find the joint pdf of $Y_1$ and $Y_2$.

(b) Determine the marginal pdf of $Y_1$ and show that $Y_1$ has an F distribution. (This is another, but equivalent, way of finding the pdf of F.)