ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5234 on D210: Gaussian random real number
Sample mean of independent gaussian random real numbers is a gaussian random real number
Formulation 0
Let $X_1 \in \text{Gaussian}(\mu_1, \sigma^2_1), \ldots, X_N \in \text{Gaussian}(\mu_N, \sigma^2_N)$ 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} \sum_{n = 1}^N \frac{X_n}{N} \overset{d}{=} \text{Gaussian} \left( \sum_{n = 1}^N \frac{\mu_n}{N}, \sum_{n = 1}^N \frac{\sigma^2_n}{N^2} \right) \end{equation}
Formulation 1
Let $X_1 \in \text{Gaussian}(\mu_1, \sigma^2_1), \ldots, X_N \in \text{Gaussian}(\mu_N, \sigma^2_N)$ 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( \sum_{n = 1}^N \frac{\mu_n}{N}, \sum_{n = 1}^N \frac{\sigma^2_n}{N^2} \right) \end{equation}
Proofs
Proof 0
Let $X_1 \in \text{Gaussian}(\mu_1, \sigma^2_1), \ldots, X_N \in \text{Gaussian}(\mu_N, \sigma^2_N)$ 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}