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 a D2713: Independent random collection |
Then
\begin{equation}
\sum_{n = 1}^N X_n
\overset{d}{=} \text{Gaussian} \left( \sum_{n = 1}^N \mu_n, \sum_{n = 1}^N \sigma^2_n \right)
\end{equation}