Square of Random Variable with t-Distribution has F-Distribution

Theorem
Let $k$ be a strictly positive integer.

Let $X \sim t_k$ where $t_k$ is the $t$-distribution with $k$ degrees of freedom.

Then:


 * $X^2 \sim F_{1, k}$

where $F_{1, k}$ is the $F$-distribution with $\tuple {1, k}$ degrees of freedom.

Proof
Let $Y \sim F_{1, k}$.

We aim to show that:


 * $\map \Pr {Y < x^2} = \map \Pr {|X| < x}$

for all $x \ge 0$.

That is:


 * $\map \Pr {Y < x^2} = \map \Pr {-x < X < x}$

for all $x \ge 0$.

We have: