Algebra of random variables
The algebra of random variables in statistics, 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 straightfoward. 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. Considering two random variables formula_1 and formula_2, the following algebraic operations are possible: In  all cases, the variable formula_8 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. The expected value formula_9 of the random variable formula_8 resulting from an algebraic operation between two random variables can be calculated using the following set of rules: If any of the random variables is replaced by a deterministic variable or by a constant  value (formula_22), the previous properties remain valid considering that formula_23 and, therefore, formula_24. If formula_8 is defined as a general non-linear algebraic function formula_26 of a random variable formula_1, then: formula_28 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 formula_1. Variance algebra for random variables. The variance formula_34 of the random variable formula_8 resulting from an algebraic operation between random variables can be calculated using the following set of rules: where formula_54 represents the covariance operator between random variables  formula_1 and formula_2. The variance of a random variable can also be expressed directly in terms of the covariance or in terms of the expected value: formula_57 If any of the random variables is replaced by a deterministic variable or by a constant value (formula_22), the previous properties remain valid considering that formula_23 and formula_24, formula_61 and formula_62. Special cases are the addition and multiplication of a random variable with a deterministic variable or a constant, where: If formula_8 is defined as a general non-linear algebraic function formula_26 of a random variable formula_1, then: formula_68 The exact value of  the variance of the non-linear function will depend on the particular probability distribution of the random variable formula_1. Covariance algebra for random variables. The covariance (formula_70) between the random variable formula_8 resulting from an algebraic operation and the random variable formula_1 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: formula_95 If any of the random variables is replaced by a deterministic variable or by a constant value ( formula_22), the previous properties remain valid considering that formula_97, formula_98 and formula_99. If formula_8  is defined as a general non-linear algebraic function formula_26of a random variable formula_1, then: formula_103 The exact value of the variance of the non-linear function will depend on the particular probability distribution of the random variable formula_1. Approximations by Taylor series expansions of moments. If the moments of a certain random variable formula_1are 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 formula_106as a Taylor series expansion of the moments, as follows: formula_107, where formula_108is the mean value  of formula_1. formula_110, where formula_111is the "n"-th moment of formula_1 about its mean. Note that by their definition, formula_113 and formula_114. The first order term always vanishes but was kept to obtain a closed form expression. Then, formula_115, where the Taylor expansion is truncated after the formula_116-th moment. Particularly for functions of normal random variables, it is possible to obtain a Taylor expansion in terms of the standard normal distribution: formula_117, where formula_118is a normal random variable, and formula_119is the standard normal distribution. Thus, formula_120, where the moments of the standard normal distribution are given by: formula_121 Similarly for  normal random variables, it is also possible to approximate the variance of the non-linear function as a Taylor series expansion as: formula_122, where formula_123, and formula_124 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: 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 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.