> [!NOTE] **Definition** (Cumulative Distribution Function 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 *cumulative distribution function* of $X$ is the function $F_{X}: \mathbb{R} \to [0,\infty)$ given by $F_{X}(a)=\mathbb{P}(\{ \omega\in \Omega : X(\omega) \leq \alpha \})$ # Properties By [[Cumulative Distribution Determines Probability Distribution for Real-Valued Random Variable]], if two real-valued random variables have the same cumulative distribution function, then they have the same probability distribution. By [[Cumulative Distribution Function of Real-Valued Random Variable is Right-Continuous]], $\lim_{ n \to \infty }F_{X}\left( x+\frac{1}{n} \right)=F_{X}(x).$