Square of Chi Random Variable has Chi-Squared Distribution

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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:

\(\ds \map \Pr {Y < x^2}\) \(=\) \(\ds \int_0^{x^2} \frac 1 {2^{n / 2} \map \Gamma {n / 2} } t^{\paren {n / 2} - 1} e^{- t / 2} \rd t\) Definition of Chi-Squared Distribution
\(\ds \) \(=\) \(\ds \frac 2 {2^{n / 2} \map \Gamma {n / 2} } \int_0^x u \paren {u^2}^{\paren {n / 2} - 1} e^{- u^2 / 2} \rd u\) substituting $t = u^2$
\(\ds \) \(=\) \(\ds \frac 1 {2^{\paren {n / 2} - 1} \map \Gamma {n / 2} } \int_0^x u u^{2 \paren {\paren {n / 2} - 1} } e^{- u^2 / 2} \rd u\)
\(\ds \) \(=\) \(\ds \frac 1 {2^{\paren {n / 2} - 1} \map \Gamma {n / 2} } \int_0^x u^{n - 1} e^{- u^2 / 2} \rd u\)
\(\ds \) \(=\) \(\ds \map \Pr {X < x}\) Definition of Chi Distribution

$\blacksquare$