Deriving the Fisher F-Distribution from Chi-Squared Variables

Let $X \sim \chi^2(m)$ and $Y \sim \chi^2(n)$ be independent. Define $$F = \frac{X/m}{Y/n}.$$

(a) Using the transformation $(F, W) = \bigl(\frac{nX}{mY},\; Y\bigr)$, compute the Jacobian and derive the joint density $f_{F,W}$.

(b) Integrate out $W$ to obtain the marginal PDF of $F$ and verify it matches the $F(m, n)$ distribution.

(c) Show that $E[F] = \dfrac{n}{n-2}$ for $n > 2$.