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:
  1. complex 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 abelian *-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
  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.

References

  • Peter Whittle, Probability via Expectation, Fourth Edition, Springer, 2000
Algebra is a branch of mathematics concerning the study of structure, relation and quantity. The name is derived from the treatise written by the Arabic[1] mathematician, astronomer, astrologer and geographer,
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axiom is a sentence or proposition that is not proved or demonstrated and is considered as self-evident or as an initial necessary consensus for a theory building or acceptation.
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Probability theory is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities
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A random variable is an abstraction of the intuitive concept of chance into the theoretical domains of mathematics, forming the foundations of probability theory and mathematical statistics.
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probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied.
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expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).
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In mathematics the concept of a measure generalizes notions such as "length", "area", and "volume" (but not all of its applications have to do with physical sizes). Informally, given some base set, a "measure" is any consistent assignment of "sizes" to (some of) the subsets of the
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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,

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In mathematics, a complex number is a number of the form


where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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Commutativity is a widely used mathematical term that refers to the ability to change the order of something without changing the end result. It is a fundamental property in most branches of mathematics and many proofs depend on it.
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Abelian, in mathematics, is used in many different definitions, named after Norwegian mathematician Niels Henrik Abel:

In group theory:
  • Abelian group, a group in which the binary operation is commutative

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*-ring

In mathematics, a *-ring is an associative ring with a map * : AA which is an antiautomorphism, and an involution.

More precisely, * is required to satisfy the following properties:
for all x
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