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.
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.
), the previous properties remain valid considering that
is defined as a general non-linear algebraic function
Some examples of this property include: The exact value of the expectation of the non-linear function will depend on the particular probability distribution of the random variable
resulting from an algebraic operation between random variables can be calculated using the following set of rules: where
represents the covariance operator between random variables
The variance of a random variable can also be expressed directly in terms of the covariance or in terms of the expected value:
is defined as a general non-linear algebraic function
The exact value of the variance of the non-linear function will depend on the particular probability distribution of the random variable
resulting from an algebraic operation and the random variable
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:
is defined as a general non-linear algebraic function
The exact value of the covariance of the non-linear function will depend on the particular probability distribution of the random variable
are known (or can be determined by integration if the probability density function is known), then it is possible to approximate the expected value of any general non-linear function
The first order term always vanishes but was kept to obtain a closed form expression.
Particularly for functions of normal random variables, it is possible to obtain a Taylor expansion in terms of the standard normal distribution:[1]
Similarly for normal random variables, it is also possible to approximate the variance of the non-linear function as a Taylor series expansion as:
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: This means that random variables form complex commutative *-algebras.
An expectation E on an algebra A of random variables is a normalized, positive linear functional.
What this means is that One may generalize this setup, allowing the algebra to be noncommutative.