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

Theorem

Let $n$ and $m$ be strictly positive integers.

Let $X$ and $Y$ be independent random variables.

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

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

Then:

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

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

Proof

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