Random variable
High school
A random variable is a number whose value depends on the outcome of a random experiment, such as the number shown by a rolled die.
Undergraduate (measure-theoretic probability)
Consider a probability space $(\Omega, \mathcal{F}, P)$. A random variable is a measurable function $X : \Omega \to \reals$. For each outcome $\omega$ of an experiment, $X$ produces a real number $X(\omega).$
Notes
Despite its name, a random variable is a function, not a variable, and it is not random: the randomness is in the outcome $\omega \in \Omega$.