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 |
$\blacksquare$