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}