> [!NOTE] Definition 1 (Support of $\mathbb{P}_{X}$ when $X$ is Discrete Real-Valued Random Variable) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a [[Discrete random variables|discrete real-valued random variable]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ Let $\mathbb{P}_{X}$ be the [[Probability Distribution of Real-Valued Random Variable|distribution]] of $X.$ The discrete support of $\mathbb{P}_{X},$ denoted $D_{X},$ is the [[Support of a real-valued function|support]] of $\mathbb{P}_{X}$ (which is the same as that of $p_{X}$): that is $D_{X}=\{ x\in \mathbb{R} : p_{X}(x)>0 \},$where $p_{X}$ denotes the [[Probability Mass Function of Discrete Real-Valued Random Variable|probability mass function]] of $X$: that is $p_{X}(x)=\mathbb{P}_{X}(\{ x \}).$ > # Applications 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.