Algebra of random variables

Algebra of random variables

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:
# complex constants are random variables;
# 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 ("ab")* = "b"* "a"* and "a"** = "a" for all random variables "a", "b", and coinciding with complex conjugation if "a" is a constant.

This means that random variables form complex commutative *-algebras. If "a" = "a"*, 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
# "E"("k") = "k" where "k" is a constant;
# "E"("a"* "a") ≥ 0 for all random variables "a";
# "E"("a" + "b") = "E"("a") + "E"("b") for all random variables "a" and "b"; and
# "E"("za") = "zE"("a") if "z" is a constant.

References

* Peter Whittle, "Probability via Expectation", Fourth Edition, Springer, 2000


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Taylor expansions for the moments of functions of random variables — In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite. This technique is often used by… …   Wikipedia

  • Random element — The term random element was introduced by Maurice Frechet in 1948 to refer to a random variable that takes values in spaces more general than had previously been widely considered. Frechet commented that the development of probability theory and… …   Wikipedia

  • Multivariate random variable — In mathematics, probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose values is unknown, either because the value has not yet occurred or because there is imperfect… …   Wikipedia

  • Information algebra — Classical information theory goes back to Claude Shannon. It is a theory of information transmission, looking at communication and storage. However, it has not been considered so far that information comes from different sources and that it is… …   Wikipedia

  • Cylindrical σ-algebra — In mathematics specifically, in measure theory and functional analysis the cylindrical σ algebra is a σ algebra often used in the study of probability measures and random variables on Banach spaces. For a Banach space X, the cylindrical σ algebra …   Wikipedia

  • History of algebra — Elementary algebra is the branch of mathematics that deals with solving for the operands of arithmetic equations. Modern or abstract algebra has its origins as an abstraction of elementary algebra. Historians know that the earliest mathematical… …   Wikipedia

  • Ratio distribution — A ratio distribution (or quotient distribution ) is a statistical distribution constructed as the distribution of the ratio of random variables having two other distributions.Given two stochastic variables X and Y , the distribution of the… …   Wikipedia

  • Probability space — This article is about mathematical term. For the novel, see Probability Space (novel). In probability theory, a probability space or a probability triple is a mathematical construct that models a real world process (or experiment ) consisting of… …   Wikipedia

  • Covariance — This article is about the measure of linear relation between random variables. For other uses, see Covariance (disambiguation). In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a …   Wikipedia

  • Glossary of probability and statistics — The following is a glossary of terms. It is not intended to be all inclusive. Concerned fields *Probability theory *Algebra of random variables (linear algebra) *Statistics *Measure theory *Estimation theory Glossary *Atomic event : another name… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”