> [!NOTE] Definition > Let $(X_{i})_{i\geq 1}$ be a [[Sequences|sequence]] of [[Pairwise Independent Set of Discrete Real-Valued Random Variables|pairwise independent discrete real-valued random variables]] that are [[Square-Integrable Discrete Real-Valued Random Variable|square-integrable]]. Then the variables are [[Uncorrelated Square-Integrable Discrete Real-Valued Random Variables|uncorrelated]]. **Proof**: By [[Expectation of Product of Two Independent Discrete Real-Valued Random Variables]] and [[Covariance of Square-Integrable Discrete Real-Valued Random Variables as Expectation of Product minus Product of Expectations|equivalent definition of covariance]], for all $j \neq k,$ we have $\text{Cov}(X_{j},X_{k})=\mathbb{E}[XY]-\mathbb{E}[X]\cdot \mathbb{E}[Y]=0$which shows that they are uncorrelated. # Applications By [[Variance of Sum of Pairwise Independent Square-Integrable Discrete Real-Valued Random Variables]], $\text{Var}\left( \sum X_{i} \right)=\sum \text{Var}(X_{i})$ when $X_{i}$ are pairwise independent.