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Wikipedia / Algebra of random variables

In statistics, 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 (or expected values), variances and covariances of such combinations.

In principle, the elementary algebra of random variables is equivalent to that of conventional non-random (or deterministic) variables. However, the changes occurring on the probability distribution of a random variable obtained after performing algebraic operations are not straightforward. Therefore, the behavior of the different operators of the probability distribution, such as expected values, variances, covariances, and moments, may be different from that observed for the random variable using symbolic algebra. It is possible to identify some key rules for each of those operators, resulting in different types of algebra for random variables, apart from the elementary symbolic algebra: Expectation algebra, Variance algebra, Covariance algebra, Moment algebra, etc.

Elementary symbolic algebra of random variables

, the following algebraic operations are possible:

resulting from each operation is also a random variable. All commutative and associative properties of conventional algebraic operations are also valid for random variables. If any of the random variables is replaced by a deterministic variable or by a constant value, all the previous properties remain valid.

Expectation algebra for random variables

resulting from an algebraic operation between two random variables can be calculated using the following set of rules:

.

, then:

Some examples of this property include:

.

Variance algebra for random variables

resulting from an algebraic operation between random variables can be calculated using the following set of rules:

represents the covariance operator.

The variance of a random variable can also be expressed directly in terms of the covariance or in terms of the expected value:

. Special cases are the addition and multiplication of a random variable with a deterministic variable or a constant, where:

, then:

.

Covariance algebra for random variables

can be calculated using the following set of rules:

The covariance of a random variable can also be expressed directly in terms of the expected value:

If any of the random variables is replaced by a deterministic variable or by a constant value the previous properties remain valid considering that and

, then:

.

Approximations by Taylor series expansions of moments

If the moments are known (or can be determined by integration if the probability density function as a Taylor series expansion of the moments, as follows:

.

is the n. The first order term always vanishes but was kept to obtain a closed form expression.

Then,

-th moment.

Particularly for functions of normal random variables, it is possible to obtain a Taylor expansion in terms of the standard normal distribution:

is the standard normal distribution. Thus,

where the moments of the standard normal distribution are given by:

Similarly for normal random variables, it is also possible to approximate the variance of the non-linear function as a Taylor series expansion as:

Algebra of complex 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 possible realizations of a random variable;

  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 and for all random variables and coinciding with complex conjugation if is a constant.

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

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

  1. where is a constant;

  2. for all random variables ;

  3. for all random variables and ; and

  4. if 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.

See also

Ratio distribution

References

Further reading