Random variables

Assume that a probability space $(S,{\cal F},P)$ is given. A random variable^9.1 $X$ is a function that maps $S$ into ${\mathbb{R}}$. Thus, $X$ assigns a real value to every element of the sample space. This enables statistics to be conveniently computed over a probability space. If $S$ is already a subset of ${\mathbb{R}}$, $X$ may by default represent the identity function.