Reciprocal of Random Variable with F-Distribution has F-Distribution
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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.
- $X = \dfrac {Y / n} {Z / m} \sim F_{n, m}$
Applying Quotient of Independent Random Variables with Chi-Squared Distribution Divided by Degrees of Freedom has F-Distribution again, we have:
- $\dfrac 1 X = \dfrac {Z / m} {Y / n} \sim F_{m, n}$
$\blacksquare$