Editing Algebra of random variables (section)

== Covariance algebra for random variables ==
The covariance (<math>\operatorname{Cov}[Z,X]</math>) between the random variable <math>Z</math> resulting from an algebraic operation and the random variable <math>X</math> can be calculated using the following set of rules:

*[[Addition]]: <math display="block">\operatorname{Cov}[Z,X] = \operatorname{Cov}[X+Y,X] = \operatorname{Var}[X] + \operatorname{Cov}[X,Y].</math> If <math>X</math> and <math>Y</math> are [[Independence (probability theory)|independent]] from each other, then: <math display="block">\operatorname{Cov}[X+Y,X] = \operatorname{Var}[X].</math>
*[[Subtraction]]: <math display="block">\operatorname{Cov}[Z,X] = \operatorname{Cov}[X-Y,X] = \operatorname{Var}[X] - \operatorname{Cov}[X,Y].</math> If <math>X</math> and <math>Y</math> are independent from each other, then: <math display="block">\operatorname{Cov}[X-Y,X] = \operatorname{Var}[X].</math>
*[[Multiplication]]: <math display="block">\operatorname{Cov}[Z,X] = \operatorname{Cov}[XY,X] = \operatorname{E}[X^2Y] - \operatorname{E}[XY] \operatorname{E}[X].</math> If <math>X</math> and <math>Y</math> are independent from each other, then: <math display="block">\operatorname{Cov}[XY,X] = \operatorname{Var}[X] \cdot \operatorname{E}[Y].</math>
*[[Division (mathematics)|Division]] (covariance with respect to the numerator): <math display="block">\operatorname{Cov}[Z,X] = \operatorname{Cov}[X/Y,X] = \operatorname{E}[X^2/Y] - \operatorname{E}[X/Y] \operatorname{E}[X].</math> If <math>X</math> and <math>Y</math> are independent from each other, then: <math display="block">\operatorname{Cov}[X/Y,X] = \operatorname{Var}[X] \cdot \operatorname{E}[1/Y].</math>
*[[Division (mathematics)|Division]] (covariance with respect to the denominator): <math display="block">\operatorname{Cov}[Z,X] = \operatorname{Cov}[Y/X,X] = \operatorname{E}[Y] - \operatorname{E}[Y/X] \operatorname{E}[X].</math> If <math>X</math> and <math>Y</math> are independent from each other, then: <math display="block">\operatorname{Cov}[Y/X,X] = \operatorname{E}[Y] \cdot (1-\operatorname{E}[X] \cdot \operatorname{E}[1/X]).</math>
*[[Exponentiation]] (covariance with respect to the base): <math display="block">\operatorname{Cov}[Z,X] = \operatorname{Cov}[X^Y,X] = \operatorname{E}[X^{Y+1}]-\operatorname{E}[X^Y] \operatorname{E}[X].</math>
*[[Exponentiation]] (covariance with respect to the power): <math display="block">\operatorname{Cov}[Z,X] = \operatorname{Cov}[Y^X,X] = \operatorname{E}[XY^X]-\operatorname{E}[Y^X] \operatorname{E}[X].</math>

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

<math display="block">\operatorname{Cov}(X,Y) = \operatorname{E}[XY] - \operatorname{E}[X]\operatorname{E}[Y]</math>

If any of the random variables is replaced by a deterministic variable or by a constant value {{nowrap|(<math>k</math>),}} the previous properties remain valid considering that {{nowrap|<math>\operatorname{E}[k] = k</math>,}} <math>\operatorname{Var}[k] = 0</math> and {{nowrap|<math>\operatorname{Cov}[X,k]=0</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{Cov}[Z,X] = \operatorname{Cov}[f(X),X] = \operatorname{E}[Xf(X)] - \operatorname{E}[f(X)] \operatorname{E}[X]</math>

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