Editing Algebra of random variables (section)

==Algebra of complex random variables==
In the [[algebra]]ic [[axiom]]atization of [[probability theory]], the primary concept is not that of probability of an event, but rather that of a [[random variable]]. [[Probability distribution]]s are determined by assigning an [[expected value|expectation]] to each random variable. The [[measure (mathematics)|measurable space]] and the probability measure arise from the random variables and expectations by means of well-known [[representation theorem]]s of analysis. One of the important features of the algebraic approach is that apparently infinite-dimensional probability distributions are not harder to formalize than finite-dimensional ones.

Random variables are assumed to have the following properties:
# [[complex number|complex]] constants are possible [[Realization (probability)|realizations]] of a random variable;
# the sum of two random variables is a random variable;
# the product of two random variables is a random variable;
# addition and multiplication of random variables are both [[commutative]]; and
# there is a notion of conjugation of random variables, satisfying {{math|1=(''XY'')<sup>*</sup> = ''Y''<sup>*</sup>''X''<sup>*</sup>}} and {{math|1=''X''<sup>**</sup> = ''X''}} for all random variables {{math|''X'',''Y''}} and coinciding with complex conjugation if {{math|''X''}} is a constant.

This means that random variables form complex commutative [[*-algebra]]s. If {{math|1=''X'' = ''X''<sup>*</sup>}} then the random variable {{math|''X''}} is called "real".

An expectation {{math|E}} on an algebra {{math|''A''}} of random variables is a normalized, positive [[linear functional]]. What this means is that
# {{math|1=E[''k''] = ''k''}} where {{math|''k''}} is a constant;
# {{math|E[''X''<sup>*</sup>''X''] ≥ 0}} for all random variables {{math|''X''}};
# {{math|1=E[''X'' + ''Y''] = E[''X''] + E[''Y'']}} for all random variables {{math|''X''}} and {{math|''Y''}}; and
# {{math|1=E[''kX''] = ''k''E[''X'']}} if {{math|''k''}} is a constant.

One may generalize this setup, allowing the algebra to be noncommutative.  This leads to other areas of noncommutative probability such as [[quantum probability]],  [[random matrix theory]], and [[free probability]].