Reciprocal of Random Variable with F-Distribution has F-Distribution

Theorem

Let $n, m$ be strictly positive integers.

Let $X \sim F_{n, m}$ where $F_{n, m}$ is the F-distribution with $\tuple {n, m}$ degrees of freedom.

Then:

$\dfrac 1 X \sim F_{m, n}$

Proof

Let $Y \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.

Let $Z \sim \chi^2_m$ where $\chi^2_m$ is the chi-squared distribution with $m$ degrees of freedom.

Then, by Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has F-Distribution:

$X = \dfrac {Y / n} {Z / m} \sim F_{n, m}$

Applying Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has F-Distribution again, we have:

$\dfrac 1 X = \dfrac {Z / m} {Y / n} \sim F_{m, n}$

$\blacksquare$