> [!NOTE] Definition 1 (Continuous Real-Valued Random Variable) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a [[Random Variable|real-valued random variable]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ Then $X$ is a continuous real-valued random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ iff the [[Cumulative Distribution Function of Real-Valued Random Variable|cumulative distribution function]] of $X$ is [[Continuous Real Function|continuous]]. > [!NOTE] Definition 2 (Continuous Real-Valued Random Variable) > Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a [[Probability Space|probability space]]. Let $X$ be a [[Random Variable|real-valued random variable]] on $(\Omega, \mathcal{F}, \mathbb{P}).$ Then $X$ is a continuous real-valued random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ if there a [[Probability Density Function|probability density function]] such that for all $a,b\in \mathbb{R},$ $\mathbb{P}(a\leq X\leq b)=\int_{a}^{b} f(x) \, dx .$ **Note**: $a\leq X\leq b$ denotes the event $\{ \omega\in \Omega: a \leq X(\omega)\leq b \}.$ # Properties