> [!NOTE] Theorem > Let $n\in \mathbb{N}^{+}$ and $(X_{n})_{i=1}^{n}$ be a list of [[Square-Integrable Discrete Real-Valued Random Variable|square-integrable discrete real-valued random variables]]. Then $\text{Var}\left( \sum_{k=1}^{n} X_{k} \right) = \sum_{k=1}^{n} \text{Var}(X_{k}) + 2 \sum_{1 \leq j <k \leq n} \text{Cov}(X_{j}, X_{k})$where $\text{Var}$ and $\text{Cov}$ denote [[Variance of Square-Integrable Discrete Real-Valued Random Variable|variance]] and [[Covariance of Square-Integrable Discrete Real-Valued Random Variables|covariance]] respectively. **Proof**: Using [[Covariance of Square Integrable Discrete Real-Valued Random Variable with Itself]] and [[Covariance Square Integrable Discrete Real-Valued Random Variables is Bilinear]], $\begin{align} \text{Var} \left( \sum_{k=1}^{n} X_{k} \right) &= \text{Cov} \left( \sum_{j=1}^{n} X_{j}, \sum_{k=1}^{n} X_{k} \right) \\ &= \sum_{j=1}^{n} \sum_{k=1}^{n} \text{Cov} (X_{j}, X_{k}) \\ &= \sum_{k=1}^{n} \text{Cov} (X_{k}, X_{k}) +\sum_{j \neq k} \text{Cov}(X_{j},X_{k}) \\ &= \sum_{k=1}^{n} \text{Var}(X_{k}) + 2 \sum_{j<k} \text{Cov}(X_{j},X_{k}) \end{align}$