Algebra of random variables
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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:
- 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* 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
- 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.
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.
- Ratio distribution
- Inverse distribution
- Product distribution
- Sum of normally distributed random variables
- List of convolutions of probability distributions
- Law of total expectation
- Law of total variance
- Law of total covariance
- Law of total cumulance
||This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (April 2013) (Learn how and when to remove this template message)|
- Whittle, Peter (2000). Probability via Expectation (4th ed.). New York, NY: Springer. ISBN 978-0-387-98955-6. Retrieved 24 September 2012.
- Springer, Melvin Dale (1979). The Algebra of Random Variables. Wiley. ISBN 0-471-01406-0. Retrieved 24 September 2012.
- Hazewinkel, Michiel, ed. (2001), "Measure algebra", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4