Definition:Random Variable
in the process of being refactored, or:
has been identified as a candidate for refactoring. In particular:
definition 2 is contained in definition 1 (it pertains only to $\R$, though)
Until this has been finished, please leave {{refactor}} in the code.
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:
- $\forall x \in \R: \left\{{\omega \in \Omega: X \left({\omega}\right) \le x}\right\} \in \Sigma$
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 half-open interval $\left\{{y \in \R: y \le x}\right\}$;
- $X^{-1} \left({\left({-\infty \,.\,.\, x}\right]}\right)$ denotes the preimage of $\left({-\infty \,.\,.\, x}\right]$.
The image $\operatorname{Im} \left({X}\right)$ of $X$ is often denoted $\Omega_X$.
The word variate is often encountered which means the same thing as random variable.
