# Definitions > [!NOTE] Definition 1 (Real-Valued Random Variable) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. A real-valued random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ is a [[Function|function]] $X:\Omega\to \mathbb{R}$ such that $X^{-1}(-\infty, \alpha]\in \mathcal{F}$ for all $\alpha\in \mathbb{R}.$ **Note**: $X^{-1}((-\infty,\alpha])$ denotes the [[Preimage (of set under a function)|preimage]] of the [[Real Interval|unbounded closed interval]] $(-\infty, \alpha]$ under $X$; or simply as $X\leq a.$ > [!NOTE] Definition (General) > A random variable on a [[Probability Space|probability space]] $(\Omega,\mathcal{F}, \mathbb{P})$ is a measurable mapping $X:\Omega \to S$, where $(S,\Sigma)$ is a measurable space. [^1] **Terminology**: - A random variable is said to be **discrete** if its **state space** (set of all possible values) is countable. > [!Example] Example > Contents # Properties The [[Probability Distribution of Real-Valued Random Variable|distribution]] of a real-valued random variable the probability measure on $\mathbb{R}$ defined by $\mathbb{P}_{X}:\mathcal{B}(\mathbb{R})\to [0,1]$ such that $\mathbb{P}_{X}(B)=\mathbb{P}(\{ \omega\in \Omega: X(\omega)\in B \}).$ By [[Probability Distribution of Real-Valued Random Variable is Probability Measure]], $(\mathbb{R}, \mathcal{B}(\mathbb{R}),\mathbb{P}_{X})$ forms a probability space. The [[Cumulative Distribution Function of Real-Valued Random Variable]] is defined by $F_{X}(a)=\mathbb{P}(X^{-1}(-\infty,a])$ for all $a\in \mathbb{R}.$ The [[Expectation of Integrable Real-Valued Random Variable]] is defined by # Reference(s) [^1]: https://proofwiki.org/wiki/Definition:Random_Variable