Editing Random variable (section)

===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.