Square of Standard Gaussian Random Variable has Chi-Squared Distribution

Theorem
Let $X \sim \Gaussian 0 1$ where $\Gaussian 0 1$ is the standard Gaussian distribution.

Then $X^2 \sim \chi^2_1$ where $\chi^2_1$ is the chi-square distribution with $1$ degree of freedom.

Proof
Let $Y \sim \chi^2_1$.

We aim to show that:


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

for all real $x \ge 0$.

We have: