In the algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

ic axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

atization of probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, the primary concept is not that of probability of an event, but rather that of a random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

. Probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

s are determined by assigning an expectation

Expected value

In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

 to each random variable. The measurable space

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 and the probability measure arise from the random variables and expectations by means of well-known representation theorem

Representation theorem

In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to a concrete structure.For example,*in algebra,...

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:

1. complex
   Complex number
   A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
   
    constants are random variables;
2. the sum of two random variables is a random variable;
3. the product of two random variables is a random variable;
4. addition and multiplication of random variables are both commutative; and
5. 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

Linear functional

In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars.  In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...

. What this means is that

1. E(k) = k where k is a constant;
2. E(a^* a) ≥ 0 for all random variables a;
3. E(a + b) = E(a) + E(b) for all random variables a and b; and
4. E(za) = zE(a) if z 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

Quantum probability

Quantum probability was developed in the 1980s as a noncommutative analog of the Kolmogorovian theory of stochastic processes. One of its aims is to clarify the mathematical foundations of quantum theory and its statistical interpretation....

, random matrix theory, and free probability

Free probability

Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products....

.

See also

• Ratio distribution
  Ratio distribution
  A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions....
  
  
• Product distribution
  Product distribution
  A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions...
  
  
• Sum of normally distributed random variables
  Sum of normally distributed random variables
  In probability theory, calculation of the sum of normally distributed random variables is an instance of the arithmetic of random variables, which can be quite complex based on the probability distributions of the random variables involved and their relationships.-Independent random variables:If X...
  
  
• Slash distribution
  Slash distribution
  In probability theory, the slash distribution is the probability distribution of a standard normal variate divided by an independent standard uniform variate...
  
  
• List of convolutions of probability distributions