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) | $N \in \text{Poisson}(\theta)$ is a D2854: Poisson random natural number |
| (iv) | $N, X_1, X_2, X_3, \ldots$ is an D2713: Independent random collection |
Then
\begin{equation}
\text{Var} \left( \sum_{n = 1}^N X_n \right)
= \theta \mathbb{E} X^2_1
\end{equation}