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 $x$ if and only if 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 $\Sigma$.
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.1$: Probability mass functions