Editing Algebra of random variables (section)

== Expectation algebra for random variables ==
The expected value <math>\operatorname{E}[Z]</math> of the random variable <math>Z</math> resulting from an algebraic operation between two random variables can be calculated using the following set of rules:

* [[Addition]]: <math>\operatorname{E}[Z] = \operatorname{E}[X+Y] = \operatorname{E}[X] + \operatorname{E}[Y] = \operatorname{E}[Y] + \operatorname{E}[X]</math>
* [[Subtraction]]: <math>\operatorname{E}[Z] = \operatorname{E}[X-Y] = \operatorname{E}[X] - \operatorname{E}[Y] = -\operatorname{E}[Y] + \operatorname{E}[X]</math>
* [[Multiplication]]: <math>\operatorname{E}[Z] = \operatorname{E}[X Y] = \operatorname{E}[YX]</math>. Particularly, if <math>X</math> and <math>Y</math> are [[Independence (probability theory)|independent]] from each other, then: <math>\operatorname{E}[X Y] = \operatorname{E}[X] \cdot \operatorname{E}[Y] = \operatorname{E}[Y] \cdot \operatorname{E}[X]</math>.
* [[Division (mathematics)|Division]]: <math>\operatorname{E}[Z] = \operatorname{E}[X/Y] = \operatorname{E}[X \cdot (1/Y)] = \operatorname{E}[(1/Y) \cdot X]</math>. Particularly, if <math>X</math> and <math>Y</math> are independent from each other, then: <math>\operatorname{E}[X/Y] = \operatorname{E}[X] \cdot \operatorname{E}[1/Y] = \operatorname{E}[1/Y] \cdot \operatorname{E}[X]</math>.
* [[Exponentiation]]: <math>\operatorname{E}[Z] = \operatorname{E}[X^Y] = \operatorname{E}[e^{Y\ln(X)}]</math>

If any of the random variables is replaced by a deterministic variable or by a constant value (<math>k</math>), the previous properties remain valid considering that <math>\Pr(X = k) = 1</math> and, therefore, <math>\operatorname{E}[X] = k</math>.

If <math>Z</math> is defined as a general non-linear algebraic function <math>f</math> of a random variable <math>X</math>, then:

<math display="block">\operatorname{E}[Z] = \operatorname{E}[f(X)] \neq f(\operatorname{E}[X])</math>

Some examples of this property include:

* <math>\operatorname{E}[X^2] \neq \operatorname{E}[X]^2</math>
* <math>\operatorname{E}[1/X] \neq 1/\operatorname{E}[X]</math>
* <math>\operatorname{E}[e^X] \neq e^{\operatorname{E}[X]}</math>
* <math>\operatorname{E}[\ln(X)] \neq \ln(\operatorname{E}[X])</math>

The exact value of the expectation of the non-linear function will depend on the particular probability distribution of the random variable <math>X</math>.