Editing Random variable (section)

==Equivalence of random variables==

There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution.

In increasing order of strength, the precise definition of these notions of equivalence is given below.

===Equality in distribution===

If the sample space is a subset of the real line, random variables ''X'' and ''Y'' are ''equal in distribution'' (denoted <math>X \stackrel{d}{=} Y</math>) if they have the same distribution functions:
:<math>\operatorname{P}(X \le x) = \operatorname{P}(Y \le x)\quad\text{for all }x.</math>

To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal [[moment generating function]]s have the same distribution. This provides, for example, a useful method of checking equality of certain functions of [[Independent and identically distributed random variables|independent, identically distributed (IID) random variables]]. However, the moment generating function exists only for distributions that have a defined [[Laplace transform]].

===Almost sure equality===

Two random variables ''X'' and ''Y'' are ''equal [[almost surely]]'' (denoted <math>X \; \stackrel{\text{a.s.}}{=} \; Y</math>) if, and only if, the probability that they are different is [[Null set|zero]]:

:<math>\operatorname{P}(X \neq Y) = 0.</math>

For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:

:<math>d_\infty(X,Y)=\operatorname{ess} \sup_\omega|X(\omega)-Y(\omega)|,</math>

where "ess sup" represents the [[essential supremum]] in the sense of [[measure theory]].

===Equality===

Finally, the two random variables ''X'' and ''Y'' are ''equal'' if they are equal as functions on their measurable space:

:<math>X(\omega)=Y(\omega)\qquad\hbox{for all }\omega.</math>

This notion is typically the least useful in probability theory because in practice and in theory, the underlying [[measure space]] of the [[Experiment (probability theory)|experiment]] is rarely explicitly characterized or even characterizable.

===Practical difference between notions of equivalence===

Since we rarely explicitly construct the probability space underlying a random variable, the difference between these notions of equivalence is somewhat subtle. Essentially, two random variables considered ''in isolation'' are "practically equivalent" if they are equal in distribution -- but once we relate them to ''other'' random variables defined on the same probability space, then they only remain "practically equivalent" if they are equal almost surely.

For example, consider the real random variables ''A'', ''B'', ''C'', and ''D'' all defined on the same probability space. Suppose that ''A'' and ''B'' are equal almost surely (<math>A \; \stackrel{\text{a.s.}}{=} \; B</math>), but ''A'' and ''C'' are only equal in distribution (<math>A \stackrel{d}{=} C</math>). Then <math> A + D \; \stackrel{\text{a.s.}}{=} \; B + D</math>, but in general <math> A + D \; \neq \; C + D</math> (not even in distribution). Similarly, we have that the expectation values <math> \mathbb{E}(AD) = \mathbb{E}(BD)</math>, but in general <math> \mathbb{E}(AD) \neq \mathbb{E}(CD)</math>. Therefore, two random variables that are equal in distribution (but not equal almost surely) can have different [[covariance|covariances]] with a third random variable.