Definition:Random Variable

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Informal Definition

A random variable is a number whose value is determined unambiguously by an experiment.


Formal Definition

Definition 1

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space, and let $\left({X, \Sigma'}\right)$ be a measurable space.


A random variable (on $\left({\Omega, \Sigma, \Pr}\right)$) is a $\Sigma \, / \, \Sigma'$-measurable mapping $f: \Omega \to X$.


Definition 2

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

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

$\forall x \in \R: \left\{{\omega \in \Omega: X \left({\omega}\right) \le x}\right\} \in \Sigma$


Definition 3

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

$\forall x \in \R: X^{-1} \left({\left({-\infty \,.\,.\, x}\right]}\right) \in \Sigma$

where:

$\left({-\infty \,.\,.\, x}\right]$ denotes the unbounded closed interval $\left\{{y \in \R: y \le x}\right\}$;
$X^{-1} \left({\left({-\infty \,.\,.\, x}\right]}\right)$ denotes the preimage of $\left({-\infty \,.\,.\, x}\right]$ under $X$.


Discrete Random Variable

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.


Continuous Random Variable

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

A continuous random variable on $\left({\Omega, \Sigma, \Pr}\right)$ is a random variable $X: \Omega \to \R$ whose cumulative distribution function is continuous for all $x \in \R$.


Also known as

The word variate is often encountered which means the same thing as random variable.


The image $\operatorname{Im} \left({X}\right)$ of $X$ is often denoted $\Omega_X$.


Sources