ThmDex – An index of mathematical definitions, results, and conjectures.
Result R2259 on D2140: Random real number variance
Variance of a finite sum of 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 D3842: Uncorrelated 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}
Since we define \begin{equation} \text{Var} \left( \sum_{n = 1}^N X_n \right) = \text{Cov} \left( \sum_{n = 1}^N X_n, \sum_{n = 1}^N X_n \right) \end{equation} then the first claim is a special case of R5113: Componentwise additivity of covariance for random real numbers. If $X_1, \ldots, X_N$ are uncorrelated, then result R3632: Uncorrelated iff covariances zero shows that $\text{Cov}(X_n, X_m) = 0$ whenever $n \neq m$. Since $\text{Cov}(X_n, X_n) = \text{Var} X_n$, the second claim follows. $\square$