Definition:Random Variable/Real-Valued

Definition
Let $\mathcal E$ be an experiment with a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

A random variable on $\left({\Omega, \Sigma, \Pr}\right)$ is a mapping $X: \Omega \to \R$ such that:
 * $\forall x \in \R: \left\{{\omega \in \Omega: X \left({\omega}\right) \le x}\right\} \in \Sigma$

The image $\operatorname{Im} \left({X}\right)$ of $X$ is often denoted $\Omega_X$.

Also see

 * Definition:Discrete Random Variable
 * Definition:Continuous Random Variable