Algebra of random variables

The algebra of random variables provides rules for the symbolic manipulation of random variables, while avoiding delving too deeply into the mathematically sophisticated ideas of probability theory. Its symbolism allows the treatment of sums, products, ratios and general functions of random variables, as well as dealing with operations such as finding the probability distributions and the expectations, variances and covariances of such combinations.

In the algebraic axiomatization of probability theory, the primary concept is not that of probability of an event, but rather that of a random variable. Probability distributions are determined by assigning an expectation to each random variable. The measurable space and the probability measure arise from the random variables and expectations by means of well-known representation theorems 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:

This means that random variables form complex commutative *-algebras. If a = a^* then the random variable a is called "real".

An expectation E on an algebra A of random variables is a normalized, positive linear functional. What this means is that

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.

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