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 $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {X, \Sigma'}$ be a measurable space.


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


Definition 2

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

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

$\forall x \in \R: \set {\omega \in \Omega: \map X \omega \le x} \in \Sigma$


Definition 3

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

$\forall x \in \R: X^{-1} \sqbrk {\hointl {-\infty} x} \in \Sigma$

where:

$\hointl {-\infty} x$ denotes the unbounded closed interval $\set {y \in \R: y \le x}$
$X^{-1} \sqbrk {\hointl {-\infty} x}$ denotes the preimage of $\hointl {-\infty} x$ under $X$.


Discrete Random Variable

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.


Continuous Random Variable

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

A continuous random variable on $\struct {\Omega, \Sigma, \Pr}$ 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 $\Img X$ of $X$ is often denoted $\Omega_X$.


Sources

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