> [!NOTE] Theorem > Let $n\in \mathbb{N}^{+}$ and $(X_{n})_{i=1}^{n}$ be a list of [[Uncorrelated Square-Integrable Discrete Real-Valued Random Variables|uncorrelated]] [[Square-Integrable Discrete Real-Valued Random Variable|square-integrable discrete real-valued random variables]]. Then the [[Variance of Square-Integrable Discrete Real-Valued Random Variable|variance]] of their sum satisfies $\text{Var}\left( \sum_{k=1}^{n} X_{k} \right) = \sum_{k=1}^{n} \text{Var}(X_{k})$ **Proof**: Follows directly from [[Variance of Sum of Square-Integrable Discrete Real-Valued Random Variables]], $\text{Var}\left( \sum_{k=1}^{n} X_{k} \right) = \sum_{k=1}^{n} \text{Var}(X_{k}) + 2 \sum_{1 \leq j <k \leq n} \text{Cov}(X_{j}, X_{k})= \sum_{k=1}^{n} \text{Var}(X_{k}).$ # Applications The statement of [[Variance of Sum of Pairwise Independent Square-Integrable Discrete Real-Valued Random Variables]] is the same.