Distribution Theory — reciprocal chi-square ratios and the F-distribution

Suppose $X$ and $Y$ are independent random variables having chi-square distributions with $m$ and $n$ degrees of freedom. Which of the following statements is/are correct?

$$\text{I. } \frac{nX}{mY} \text{ follows } F_{m,n}.$$

$$\text{II. } \frac{mY}{nX} \text{ follows } F_{n,m}.$$