> [!NOTE] Definition 1 (Independent Discrete Real-Valued Random Variables) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X,Y$ be [[Discrete random variables|discrete real-valued random variables]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ Let $p_{X,Y}$ denote the [[Joint Probability Mass Function of Discrete Real-Valued Random Variables|joint probability mass function]] of $X$ and $Y.$ Let $p_{X}$ denote the [[Probability Mass Function of Discrete Real-Valued Random Variable|probability mass function]] of $X.$ Then $X$ and $Y$ are independent iff for all $x,y\in \mathbb{R},$ $p_{X,Y}(x,y)=p_{X}(x)\cdot p_{Y}(y)$that is, $\mathbb{P}(X=x,Y=y)=\mathbb{P}(X=x)\cdot \mathbb{P}(Y=y).$ > **Note**: that is, the events $X^{-1}(x)$ and $Y^{-1}(y)$ are [[Independence of Two Events|independent events]]. # Properties By [[Expectation of Product of Two Independent Discrete Real-Valued Random Variables]], if $X$ and $Y$ are independent then $\mathbb{E}[XY]=\mathbb{E}[X]\cdot \mathbb{E}[Y].$ # Applications **Generalisations**: A set of discrete real-valued random variables is [[Pairwise Independent Set of Discrete Real-Valued Random Variables|pairwise independent]] iff each pair of different random variables chosen from the set are independent. The set is [[Mutually Independent Set of Discrete Real-Valued Random Variables|mutually independent]] iff for all subsets of the set, the joint probability mass function of the random variables is the product of their individual probability mass functions.