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

Finite sum of uncorrelated identically distributed 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 D3842: Uncorrelated random collection

Then \begin{equation} \sum_{n = 1}^N X_n \overset{d}{=} \text{Gaussian} \left( N \mu, N \sigma^2 \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 D3842: Uncorrelated random collection