Variance of Random Sample from Gaussian Distribution has Chi-Squared Distribution

Theorem

Let $X_1, X_2, \ldots, X_n$ form a random sample of size $n$ from the Gaussian distribution $\Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{>0}$.

Let:

$\ds \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$

and:

$\ds s^2 = \frac 1 {n - 1} \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2$

Then:

$\dfrac {\paren {n - 1} s^2} {\sigma^2} \sim \chi^2_{n - 1}$

where $\chi^2_{n - 1}$ is the chi-squared distribution with $n - 1$ degrees of freedom.

Proof

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