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

Sample mean of I.I.D. gaussian random real numbers is a gaussian random real number

Formulation 0

Let $X_1, \ldots, X_N \in \text{Gaussian}(\mu, \sigma^2)$ each be a D210: Gaussian random real number such that

(i)	$X_1, \ldots, X_N$ is an D2713: Independent random collection
(ii)	\begin{equation} \overline{X}_N : = \frac{1}{N} \sum_{n = 1}^N X_n \end{equation}

Then \begin{equation} \overline{X}_N \overset{d}{=} \text{Gaussian} \left( \mu, \frac{\sigma^2}{N} \right) \end{equation}

Formulation 1

Let $X_1, \ldots, X_N \in \text{Gaussian}(\mu, \sigma^2)$ each be a D210: Gaussian random real number such that

(i)	$X_1, \ldots, X_N$ is an D2713: Independent random collection

Then \begin{equation} \frac{1}{N} \sum_{n = 1}^N X_n \overset{d}{=} \text{Gaussian} \left( \mu, \frac{\sigma^2}{N} \right) \end{equation}

Proofs

Proof 0

Let $X_1, \ldots, X_N \in \text{Gaussian}(\mu, \sigma^2)$ each be a D210: Gaussian random real number such that

(i)	$X_1, \ldots, X_N$ is an D2713: Independent random collection
(ii)	\begin{equation} \overline{X}_N : = \frac{1}{N} \sum_{n = 1}^N X_n \end{equation}