Definition:Random Variable/Discrete

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

A discrete random variable on $\struct {\Omega, \Sigma, \Pr}$ is a mapping $X: \Omega \to \R$ such that:
 * $(1): \quad$ The image of $X$ is a countable subset of $\R$
 * $(2): \quad$ $\forall x \in \R: \set {\omega \in \Omega: \map X \omega = x} \in \Sigma$

Alternatively (and meaning exactly the same thing), the second condition can be written as:
 * $(2)': \quad$ $\forall x \in \R: \map {X^{-1} } x \in \Sigma$

where $\map {X^{-1} } x$ denotes the preimage of $x$.

Note that if $x \in \R$ is not the image of any elementary event $\omega$, then $\map {X^{-1} } x = \O$ and of course by definition of event space as a sigma-algebra, $\O \in \Sigma$.

Note that a discrete random variable also fulfils the conditions for it to be a random variable.

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

Discussion
The meaning of condition $(2)$ in this context can be explained as follows:

Suppose $X$ is a discrete random variable. Then it takes values in $\R$. But we don't know what the actual value of $X$ is going to be, since the outcome of $\EE$ involves chance.

What we can do, though, is determine the probability that $X$ takes any particular value $x$.

To do this, we note that $X$ has the value $x$ the outcome of $\EE$ lies in the subset of $\Omega$ which is mapped to $x$.

But for any such element $x$ of the image of $X$, the preimage of $x$ is an element of $\Sigma$.