Editing Random variable (section)

==Measure-theoretic definition==

The most formal, [[axiomatic]] definition of a random variable involves [[measure theory]]. Continuous random variables are defined in terms of [[set (mathematics)|set]]s of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the [[Banach–Tarski paradox]]) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a [[sigma-algebra]] to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the [[Borel σ-algebra]], which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or [[countably infinite]] number of [[union (set theory)|union]]s and/or [[intersection (set theory)|intersection]]s of such intervals.<ref name="UCSB">{{cite web|last=Steigerwald|first=Douglas G.|title=Economics 245A – Introduction to Measure Theory|url=http://faculty.econ.ucsb.edu/~doug/245a/Lectures/Measure%20Theory.pdf|access-date=April 26, 2013|publisher=University of California, Santa Barbara}}</ref>

The measure-theoretic definition is as follows.

Let <math>(\Omega, \mathcal{F}, P)</math> be a [[probability space]] and <math>(E, \mathcal{E})</math> a [[measurable space]]. Then an '''<math>(E, \mathcal{E})</math>-valued random variable''' is a measurable function <math>X\colon \Omega \to E</math>, which means that, for every subset <math>B\in\mathcal{E}</math>, its [[preimage]] is <math>\mathcal{F}</math>-measurable; <math>X^{-1}(B)\in \mathcal{F}</math>, where <math>X^{-1}(B) = \{\omega : X(\omega)\in B\}</math>.<ref>{{harvtxt|Fristedt|Gray|1996|loc=page 11}}</ref> This definition enables us to measure any subset <math>B\in \mathcal{E}</math> in the target space by looking at its preimage, which by assumption is measurable.

In more intuitive terms, a member of <math>\Omega</math> is a possible outcome, a member of <math>\mathcal{F}</math> is a measurable subset of possible outcomes, the function <math>P</math> gives the probability of each such measurable subset, <math>E</math> represents the set of values that the random variable can take (such as the set of real numbers), and a member of <math>\mathcal{E}</math> is a "well-behaved" (measurable) subset of <math>E</math> (those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a well-defined probability.

When <math>E</math> is a [[topological space]], then the most common choice for the [[σ-algebra]] <math>\mathcal{E}</math> is the [[Borel σ-algebra]] <math>\mathcal{B}(E)</math>, which is the σ-algebra generated by the collection of all open sets in <math>E</math>. In such case the <math>(E, \mathcal{E})</math>-valued random variable is called an '''<math>E</math>-valued random variable'''. Moreover, when the space <math>E</math> is the real line <math>\mathbb{R}</math>, then such a real-valued random variable is called simply a '''random variable'''.

===Real-valued random variables===

In this case the observation space is the set of real numbers. Recall,  <math>(\Omega, \mathcal{F}, P)</math> is the probability space. For a real observation space, the function <math>X\colon \Omega \rightarrow \mathbb{R}</math> is a real-valued random variable if
:<math>\{ \omega : X(\omega) \le r \} \in \mathcal{F} \qquad \forall r \in \mathbb{R}.</math>

This definition is a special case of the above because the set <math>\{(-\infty, r]: r \in \R\}</math> generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that <math>\{ \omega : X(\omega) \le r \} = X^{-1}((-\infty, r])</math>.