Definition:Random Variable/Real-Valued

Definition
Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.

A random variable on $\struct {\Omega, \Sigma, \Pr}$ is a mapping $X: \Omega \to \R$ such that:
 * $\forall x \in \R: \set {\omega \in \Omega: \map X \omega \le x} \in \Sigma$

The image $\Img X$ of $X$ is often denoted $\Omega_X$.

Also see

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