ThmDex – An index of mathematical definitions, results, and conjectures.

Bessel-corrected sample variance of independent standard gaussians is a transformed chi-squared random real number

Formulation 0

Let $Z_1, \ldots, Z_N \in \text{Gaussian}(0, 1)$ each be a D211: Standard gaussian random real number such that

(i)	\begin{equation} N \in \{ 2, 3, 4, \ldots \} \end{equation}
(ii)	$Z_1, \ldots, Z_N$ is an D2713: Independent random collection
(iii)	\begin{equation} S : = \left( \frac{1}{N - 1} \sum_{n = 1}^N Z^2_n \right)^{1 / 2} \end{equation}
(iv)	$\chi \in \text{ChiSquared}(N)$ is a D212: Chi-squared random unsigned real number

Then \begin{equation} S \overset{d}{=} \left( \frac{1}{N - 1} \chi \right)^{1 / 2} \end{equation}

Formulation 1

Let $Z_1, \ldots, Z_N \in \text{Gaussian}(0, 1)$ each be a D211: Standard gaussian random real number such that

(i)	\begin{equation} N \in \{ 2, 3, 4, \ldots \} \end{equation}
(ii)	$Z_1, \ldots, Z_N$ is an D2713: Independent random collection
(iii)	\begin{equation} S : = \sqrt{ \frac{1}{N - 1} \sum_{n = 1}^N Z^2_n } \end{equation}
(iv)	$\chi \in \text{ChiSquared}(N)$ is a D212: Chi-squared random unsigned real number

Then \begin{equation} S \overset{d}{=} \sqrt{ \frac{1}{N - 1} \chi } \end{equation}

Proofs

Proof 0

Let $Z_1, \ldots, Z_N \in \text{Gaussian}(0, 1)$ each be a D211: Standard gaussian random real number such that

(i)	\begin{equation} N \in \{ 2, 3, 4, \ldots \} \end{equation}
(ii)	$Z_1, \ldots, Z_N$ is an D2713: Independent random collection
(iii)	\begin{equation} S : = \left( \frac{1}{N - 1} \sum_{n = 1}^N Z^2_n \right)^{1 / 2} \end{equation}
(iv)	$\chi \in \text{ChiSquared}(N)$ is a D212: Chi-squared random unsigned real number

By definition, we have \begin{equation} \chi \overset{d}{=} \sum_{n = 1}^N Z^2_n \end{equation} whence the result follows. $\square$