> [!NOTE] Definition 1 (Distribution of Real-Valued Random Variable) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a [[Random Variables|real-valued random variable]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ The probability distribution of $X,$ is given by $\mathbb{P}_{X}:\mathcal{B}(\mathbb{R})\to [0,1],$ $\mathbb{P}_{X}(B)=\mathbb{P}(\{ \omega \in \Omega : X(\omega)\in B \}),$where $\mathcal{B}(\mathbb{R})$ is a [[Borel Set|Borel set]] (the smallest event space containing the intervals $(-\infty,\alpha]$ for all $\alpha\in \mathbb{R}$). # Properties By [[Probability Distribution of Real-Valued Random Variable is Probability Measure]], $\mathbb{P}_{X}$ is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R})),$ where $\mathcal{B}(\mathbb{R})$ is the smallest interval containing $(-\infty,a]$ for all $a\in \mathbb{R}.$ **Discrete distributions**: A distribution is said to be [[Discrete random variables|discrete]] if its support is a countable set. 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.