Let $X_1, X_2, X_3, \ldots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3, \ldots$ is an D3357: Identically distributed random collection
(ii)	$$\begin{equation} \mathbb{E} |X_1|^2 < \infty \end{equation}$$
(iii)	$N \in \text{Random}(\mathbb{N})$ is a D5216: Random natural number
(iv)	$$\begin{equation} \mathbb{E} N^2 < \infty \end{equation}$$
(v)	$N, X_1, X_2, X_3, \ldots$ is an D2713: Independent random collection

Then $$\begin{equation} \text{Var} \left( \sum_{n = 1}^N X_n \right) = \mathbb{E} (N) \text{Var} X_1 + \text{Var} (N) (\mathbb{E} X_1)^2 \end{equation}$$