> [!NOTE] Definition (Pairwise Independent Set of Discrete Real-Valued Random Variables) > Let $(\Omega,\mathcal{F},\mathbb{P})$ be a [[Probability Space|probability space]]. Let $(X_{i})_{i \geq 1}$ be a [[Sequences|sequence]] of [[Discrete random variables|discrete real-valued random variables]] on $(\Omega,\mathcal{F},\mathbb{P}).$ The random variables are pairwise independent iff $X_{i}$ and $X_{j}$ are [[Independence of Two Discrete Real-Valued Random Variables|independent]] for all $i \neq j$: that is, for all $x,y\in \mathbb{R},$ $p_{X_{i},X_{j}}(x,y)=p_{X_{i}}(x)\cdot p_{X_{j}}(y),$ where $p_{X,Y}$ denotes the [[Joint Probability Mass Function of Discrete Real-Valued Random Variables|joint probability mass function]] of $X$ and $Y$ and $p_{X}$ denotes the [[Probability Mass Function of Discrete Real-Valued Random Variable|probability mass function]] of $X.$ # Properties By [[Variance of Sum of Pairwise Independent Square-Integrable Discrete Real-Valued Random Variables]], if $X_{1},X_{2},\dots,X_{n}$ are pairwise independent then $\text{Var}\left( \sum_{i=1}^{n} X_{i} \right)=\sum_{i=1}^{n}\text{Var}(X_{i}).$ By [[Strong Law of Large Numbers]], ...