> [!NOTE] Definition (Mutually Independent Set of Discrete Real-Valued Random Variables) > Let $(\Omega,\mathcal{F},\mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a [[Countable Set|countable set]] of [[Discrete random variables|discrete real-valued random variables]] on $(\Omega,\mathcal{F},\mathbb{P}).$ The random variables are mutually independent iff for all [[Finite Set|finite]] [[Subsets|subsets]] $J\subset X,$ the [[Joint Probability Mass Function of Discrete Real-Valued Random Variables|joint probability mass function]] of $J$ is the product of the [[Probability Mass Function of Discrete Real-Valued Random Variable|probability mass functions]] of the individual variables: that is, for all $k\in \mathbb{N}^{+},$ $X_{1},X_{2},\dots,X_{k}\in J,$ and $(x_{1},x_{2},\dots,x_{k})\in \mathbb{R}^{k},$ $\mathbb{P}(X_{1}=x_{1},X_{2}=x_{2},\dots,X_{k}=x_{k})=\mathbb{P}(X_{1}=x_{1})\cdot \mathbb{P}(X_{2}=x_{2})\cdots \mathbb{P}(X_{k}=x_{k}).$ # Applications Two discrete real-valued random variables are [[Independence of Two Discrete Real-Valued Random Variables|independent]] iff they are mutually independent: that is, $\mathbb{P}(X=x,Y=y)=\mathbb{P}(X=x)\cdot \mathbb{P}(X=y).$