If $(\Omega,\mc{A},P)$ is a probability space, then a random variable on $\Omega$ is a measurable function $X: (\Omega,\mc{A}) \to S$ to a measurable space $S$ (frequently taken to be the real numbers with the standard measure). The law of a random variable is the probability measure $PX^{-1}:S\to \R$ defined by $PX^{-1}(s)=P(X^{-1}(s))$

A random variable $X$ is said to be discrete if the set $ \{X(\omega) : \omega \in \Omega \}$ (i.e. the range of $X$ is finite or countable. A more general version of this definition is as follows: A random variable $X$ is discrete if there is a countable subset $B$ of the range of $X$ such that $P(X \in B)=1$ (Note that, as a countable subset of $\mathbb{R}$ $B$ is measurable).

A random variable $Y$ is said to be continuous if it has a cumulative distribution function which is absolutely continuous.

Example:

Consider the event of throwing a coin. Thus, $\Omega = \{ H, T \}$ where $H$ is the event in which the coin falls head and $T$ the event in which falls tails. Let $X=$ of tails in the experiment. Then $X$ is a (discrete) random variable.