random variable | planetmath.org

If $(\Omega,\mathcal{A},P)$ is a probability space, then a random variable on $\Omega$ is a measurable function $X:(\Omega,\mathcal{A})\to S$ 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 $PX^{{-1}}:S\to\mathbb{R}$ 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.

Read full article from random variable | planetmath.org