ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5233 on D210: Gaussian random real number
Subresult of R5237: Finite sum of uncorrelated identically distributed gaussian random real numbers is a gaussian random real number
Finite sum 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 a D2713: Independent 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 a D2713: Independent random collection