Definition:Random Variable
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
- For a video presentation of the contents of this page, visit the Khan Academy.
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and check that it links to the appropriate page.
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- 2001: Michael A. Bean: Probability: The Science of Uncertainty: $\S 2.1$