Square of Chi Random Variable has Chi-Squared Distribution

Theorem
Let $n$ be a strictly positive integer.

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

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

Proof
Let $Y \sim \chi^2_n$.

We aim to show that:


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

for all $x \in \hointr 0 \infty$.

We have: