ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4687 on D2140: Random real number variance
Subresult of R2259: Variance of a finite sum of random real numbers
Additivity of variance for a finite number of independent random real numbers
Formulation 0
Let $X_1, \ldots, X_N \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that
(i) \begin{equation} \mathbb{E} |X_1|^2, \ldots, \mathbb{E} |X_N|^2 < \infty \end{equation}
Then
(1) \begin{equation} \text{Var} \left( \sum_{n = 1}^N X_n \right) = \sum_{n = 1}^N \sum_{m = 1}^N \text{Cov}(X_n, X_m) \end{equation}
(2) $\text{Var} \left( \sum_{n = 1}^N X_n \right) = \sum_{n = 1}^N \text{Var}(X_n)$ if $X_1, \ldots, X_N$ is an D2713: Independent random collection on $P$
Proofs
Proof 0
Let $X_1, \ldots, X_N \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that
(i) \begin{equation} \mathbb{E} |X_1|^2, \ldots, \mathbb{E} |X_N|^2 < \infty \end{equation}