> [!NOTE] Definition > Let $(X_{i})_{i\geq 1}$ be a [[Sequences|sequence]] of [[Square-Integrable Discrete Real-Valued Random Variable|square-integrable discrete real-valued random variables]]. The variables are uncorrelated iff for all $j,k\in \mathbb{N}^{+}$ such that $j \neq k,$ $\text{Cov}(X_{j}, X_{k}) = 0$that is, the [[Covariance of Square-Integrable Discrete Real-Valued Random Variables|covariance]] of $X_{j}$ and $X_{k}$ is zero. # Properties By [[Variance of Sum of Uncorrelated Square-Integrable Discrete Real-Valued Random Variables]], $\text{Var} \left( \sum X_{i} \right) = \sum \text{Var}(X_{i})$ if the $X_{i}$ are uncorrelated. This gives the [[Square Root Law]] which asserts that $\text{Var}\left( \frac{X_{1}+\dots X_{n}}{n} \right)=\frac{\sigma^{2}}{n}$ when the $X_{i}$ are uncorrelated and have the same variance $\sigma^{2}.$ # Applications Note that [[Pairwise Independent Square-Integrable Discrete Real-Valued Random Variables are Uncorrelated]].