Definition:Random Variable/Discrete

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Definition

Let $\mathcal E$ be an experiment with a probability space $\left({\Omega, \Sigma, \Pr}\right)$.


A discrete random variable on $\left({\Omega, \Sigma, \Pr}\right)$ is a mapping $X: \Omega \to \R$ such that:

$(1): \quad$ The image of $X$ is a countable subset of $\R$
$(2): \quad$ $\forall x \in \R: \left\{{\omega \in \Omega: X \left({\omega}\right) = x}\right\} \in \Sigma$


Alternatively (and meaning exactly the same thing), the second condition can be written as:

$(2)': \quad$ $\forall x \in \R: X^{-1} \left({x}\right) \in \Sigma$

where $X^{-1} \left({x}\right)$ denotes the preimage of $x$.


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


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


Also known as

The image $\operatorname{Im} \left({X}\right)$ 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 $\mathcal E$ 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 $\mathcal E$ 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