> [!NOTE] Definition 1 (PMF of 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 $(\Omega, \mathcal{F}, \mathbb{P}).$ The probability mass function of $X$ is given by $p_{X}:\mathbb{R}\to[0,1]$ defined by $p_{X}(x)=\mathbb{P}(X=x).$ **Note**: $X=x$ denotes $\{ \omega: X(\omega)=x \}=X^{-1}(x).$ # Properties # 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.