> [!NOTE] Definition 1 (Discrete Real-Valued Random Variable) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a [[Random Variable|real-valued random variable]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ Let $\mathbb{P}_{X}$ denote the [[Probability Distribution of Real-Valued Random Variable|distribution]] of $X.$ > > Then $\mathbb{P}_{X}$ (or $X$) is discrete iff there exists a [[Countable Set|countable set]] $S \subset \mathbb{R}$ such that $\mathbb{P}_{X}(S)=1$: that is, the [[Support of a real-valued function|support]] of $\mathbb{P}_{X}$ is countable. # Properties The [[Probability Mass Function of Discrete Real-Valued Random Variable|probability mass function]] of $X,$ denoted $p_{X},$ is defined on $\mathbb{R}$ to the be probability of each outcome: that is $p_{X}(x)=\mathbb{P}_{X}(\{ x \}),$ where $\mathbb{P}_{X}$ denotes the distribution of $X.$ The set $S$ is the [[Discrete Support of Distribution of Discrete Real-Valued Random Variable|discrete support]] of $\mathbb{P}_{X}.$ By [[Probability Distribution of Discrete Real-Valued Random Variable in Terms of Probability Mass Function]], $\mathbb{P}_{X}(B) =\sum_{x\in B \cap D_{X}} p_{X}(x).$ Thus in order to understand the distribution of $X,$ it is sufficient to understand its mass function. A discrete real-valued random variable is [[Square-Integrable Discrete Real-Valued Random Variable|square-integrable]] iff ...