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Concept
Algebra of random variables
Résumé
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 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.
Considering two random variables and , the following algebraic operations are possible:
Addition:
Subtraction:
Multiplication:
Division:
Exponentiation:
In all cases, the variable 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.
The expected value of the random variable resulting from an algebraic operation between two random variables can be calculated using the following set of rules:
Addition:
Subtraction:
Multiplication: . Particularly, if and are independent from each other, then: .
Division: .
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