Algebra of random variables

<h2 id="elementary-symbolic-algebra-of-random-variables">Elementary symbolic algebra of random variables</h2>
Considering two random variables 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
, the following algebraic operations are possible:

<ul><li><a href="/facts/Addition/apFWJBFB">Addition</a>: 
 
 
 
 Z
 =
 X
 +
 Y
 =
 Y
 +
 X
 
 
 {\displaystyle Z=X+Y=Y+X}
 
</li>
<li><a href="/facts/Subtraction/dEUrp6k0">Subtraction</a>: 
 
 
 
 Z
 =
 X
 −
 Y
 =
 −
 Y
 +
 X
 
 
 {\displaystyle Z=X-Y=-Y+X}
 
</li>
<li><a href="/facts/Multiplication/VtcUjpNv">Multiplication</a>: 
 
 
 
 Z
 =
 X
 Y
 =
 Y
 X
 
 
 {\displaystyle Z=XY=YX}
 
</li>
<li><a href="/facts/Division_(mathematics)/FqLKNkmq">Division</a>: Suppose 
 
 
 
 Y
 ≠
 0
 
 
 {\displaystyle Y\neq 0}
 
, 
 
 
 
 Z
 =
 X
 
 /
 
 Y
 =
 X
 ⋅
 (
 1
 
 /
 
 Y
 )
 =
 (
 1
 
 /
 
 Y
 )
 ⋅
 X
 
 
 {\displaystyle Z=X/Y=X\cdot (1/Y)=(1/Y)\cdot X}
 
.</li>
<li><a href="/facts/Exponentiation/pac6QY2g">Exponentiation</a>: 
 
 
 
 Z
 =
 
 X
 
 Y
 
 
 =
 
 e
 
 Y
 ln
 ⁡
 (
 X
 )
 
 
 
 
 {\displaystyle Z=X^{Y}=e^{Y\ln(X)}}
 
</li></ul>
In all cases, the variable 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 resulting from each operation is also a random variable. All <a href="/facts/Commutative_property/WaYxJLxd">commutative</a> and <a href="/facts/Associative_property/EQX8lV6r">associative</a> 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.

<h2 id="expectation-algebra-for-random-variables">Expectation algebra for random variables</h2>
The expected value 
 
 
 
 E
 ⁡
 [
 Z
 ]
 
 
 {\displaystyle \operatorname {E} [Z]}
 
 of the random variable 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 resulting from an algebraic operation between two random variables can be calculated using the following set of rules:

<ul><li><a href="/facts/Addition/apFWJBFB">Addition</a>: 
 
 
 
 E
 ⁡
 [
 Z
 ]
 =
 E
 ⁡
 [
 X
 +
 Y
 ]
 =
 E
 ⁡
 [
 X
 ]
 +
 E
 ⁡
 [
 Y
 ]
 =
 E
 ⁡
 [
 Y
 ]
 +
 E
 ⁡
 [
 X
 ]
 
 
 {\displaystyle \operatorname {E} [Z]=\operatorname {E} [X+Y]=\operatorname {E} [X]+\operatorname {E} [Y]=\operatorname {E} [Y]+\operatorname {E} [X]}
 
</li>
<li><a href="/facts/Subtraction/dEUrp6k0">Subtraction</a>: 
 
 
 
 E
 ⁡
 [
 Z
 ]
 =
 E
 ⁡
 [
 X
 −
 Y
 ]
 =
 E
 ⁡
 [
 X
 ]
 −
 E
 ⁡
 [
 Y
 ]
 =
 −
 E
 ⁡
 [
 Y
 ]
 +
 E
 ⁡
 [
 X
 ]
 
 
 {\displaystyle \operatorname {E} [Z]=\operatorname {E} [X-Y]=\operatorname {E} [X]-\operatorname {E} [Y]=-\operatorname {E} [Y]+\operatorname {E} [X]}
 
</li>
<li><a href="/facts/Multiplication/VtcUjpNv">Multiplication</a>: 
 
 
 
 E
 ⁡
 [
 Z
 ]
 =
 E
 ⁡
 [
 X
 Y
 ]
 =
 E
 ⁡
 [
 Y
 X
 ]
 
 
 {\displaystyle \operatorname {E} [Z]=\operatorname {E} [XY]=\operatorname {E} [YX]}
 
. Particularly, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are <a href="/facts/Independence_(probability_theory)/NUzQtnUL">independent</a> from each other, then: 
 
 
 
 E
 ⁡
 [
 X
 Y
 ]
 =
 E
 ⁡
 [
 X
 ]
 ⋅
 E
 ⁡
 [
 Y
 ]
 =
 E
 ⁡
 [
 Y
 ]
 ⋅
 E
 ⁡
 [
 X
 ]
 
 
 {\displaystyle \operatorname {E} [XY]=\operatorname {E} [X]\cdot \operatorname {E} [Y]=\operatorname {E} [Y]\cdot \operatorname {E} [X]}
 
.</li>
<li><a href="/facts/Division_(mathematics)/FqLKNkmq">Division</a>: 
 
 
 
 E
 ⁡
 [
 Z
 ]
 =
 E
 ⁡
 [
 X
 
 /
 
 Y
 ]
 =
 E
 ⁡
 [
 X
 ⋅
 (
 1
 
 /
 
 Y
 )
 ]
 =
 E
 ⁡
 [
 (
 1
 
 /
 
 Y
 )
 ⋅
 X
 ]
 
 
 {\displaystyle \operatorname {E} [Z]=\operatorname {E} [X/Y]=\operatorname {E} [X\cdot (1/Y)]=\operatorname {E} [(1/Y)\cdot X]}
 
. Particularly, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are independent from each other, then: 
 
 
 
 E
 ⁡
 [
 X
 
 /
 
 Y
 ]
 =
 E
 ⁡
 [
 X
 ]
 ⋅
 E
 ⁡
 [
 1
 
 /
 
 Y
 ]
 =
 E
 ⁡
 [
 1
 
 /
 
 Y
 ]
 ⋅
 E
 ⁡
 [
 X
 ]
 
 
 {\displaystyle \operatorname {E} [X/Y]=\operatorname {E} [X]\cdot \operatorname {E} [1/Y]=\operatorname {E} [1/Y]\cdot \operatorname {E} [X]}
 
.</li>
<li><a href="/facts/Exponentiation/pac6QY2g">Exponentiation</a>: 
 
 
 
 E
 ⁡
 [
 Z
 ]
 =
 E
 ⁡
 [
 
 X
 
 Y
 
 
 ]
 =
 E
 ⁡
 [
 
 e
 
 Y
 ln
 ⁡
 (
 X
 )
 
 
 ]
 
 
 {\displaystyle \operatorname {E} [Z]=\operatorname {E} [X^{Y}]=\operatorname {E} [e^{Y\ln(X)}]}
 
</li></ul>
If any of the random variables is replaced by a deterministic variable or by a constant value (
 
 
 
 k
 
 
 {\displaystyle k}
 
), the previous properties remain valid considering that 
 
 
 
 Pr
 (
 X
 =
 k
 )
 =
 1
 
 
 {\displaystyle \Pr(X=k)=1}
 
 and, therefore, 
 
 
 
 E
 ⁡
 [
 X
 ]
 =
 k
 
 
 {\displaystyle \operatorname {E} [X]=k}
 
.
If 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 is defined as a general non-linear algebraic function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 of a random variable 
 
 
 
 X
 
 
 {\displaystyle X}
 
, then:

 
 
 
 E
 ⁡
 [
 Z
 ]
 =
 E
 ⁡
 [
 f
 (
 X
 )
 ]
 ≠
 f
 (
 E
 ⁡
 [
 X
 ]
 )
 
 
 {\displaystyle \operatorname {E} [Z]=\operatorname {E} [f(X)]\neq f(\operatorname {E} [X])}

Some examples of this property include:

<ul><li>
 
 
 
 E
 ⁡
 [
 
 X
 
 2
 
 
 ]
 ≠
 E
 ⁡
 [
 X
 
 ]
 
 2
 
 
 
 
 {\displaystyle \operatorname {E} [X^{2}]\neq \operatorname {E} [X]^{2}}
 
</li>
<li>
 
 
 
 E
 ⁡
 [
 1
 
 /
 
 X
 ]
 ≠
 1
 
 /
 
 E
 ⁡
 [
 X
 ]
 
 
 {\displaystyle \operatorname {E} [1/X]\neq 1/\operatorname {E} [X]}
 
</li>
<li>
 
 
 
 E
 ⁡
 [
 
 e
 
 X
 
 
 ]
 ≠
 
 e
 
 E
 ⁡
 [
 X
 ]
 
 
 
 
 {\displaystyle \operatorname {E} [e^{X}]\neq e^{\operatorname {E} [X]}}
 
</li>
<li>
 
 
 
 E
 ⁡
 [
 ln
 ⁡
 (
 X
 )
 ]
 ≠
 ln
 ⁡
 (
 E
 ⁡
 [
 X
 ]
 )
 
 
 {\displaystyle \operatorname {E} [\ln(X)]\neq \ln(\operatorname {E} [X])}
 
</li></ul>
The exact value of the expectation of the non-linear function will depend on the particular probability distribution of the random variable 
 
 
 
 X
 
 
 {\displaystyle X}
 
.

<h2 id="variance-algebra-for-random-variables">Variance algebra for random variables</h2>
The variance 
 
 
 
 Var
 ⁡
 [
 Z
 ]
 
 
 {\displaystyle \operatorname {Var} [Z]}
 
 of the random variable 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 resulting from an algebraic operation between random variables can be calculated using the following set of rules:

<ul><li><a href="/facts/Addition/apFWJBFB">Addition</a>: 
 
 
 
 Var
 ⁡
 [
 Z
 ]
 =
 Var
 ⁡
 [
 X
 +
 Y
 ]
 =
 Var
 ⁡
 [
 X
 ]
 +
 2
 Cov
 ⁡
 [
 X
 ,
 Y
 ]
 +
 Var
 ⁡
 [
 Y
 ]
 .
 
 
 {\displaystyle \operatorname {Var} [Z]=\operatorname {Var} [X+Y]=\operatorname {Var} [X]+2\operatorname {Cov} [X,Y]+\operatorname {Var} [Y].}
 
Particularly, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are <a href="/facts/Independence_(probability_theory)/NUzQtnUL">independent</a> from each other, then: 
 
 
 
 Var
 ⁡
 [
 X
 +
 Y
 ]
 =
 Var
 ⁡
 [
 X
 ]
 +
 Var
 ⁡
 [
 Y
 ]
 .
 
 
 {\displaystyle \operatorname {Var} [X+Y]=\operatorname {Var} [X]+\operatorname {Var} [Y].}
 
</li>
<li><a href="/facts/Subtraction/dEUrp6k0">Subtraction</a>: 
 
 
 
 Var
 ⁡
 [
 Z
 ]
 =
 Var
 ⁡
 [
 X
 −
 Y
 ]
 =
 Var
 ⁡
 [
 X
 ]
 −
 2
 Cov
 ⁡
 [
 X
 ,
 Y
 ]
 +
 Var
 ⁡
 [
 Y
 ]
 .
 
 
 {\displaystyle \operatorname {Var} [Z]=\operatorname {Var} [X-Y]=\operatorname {Var} [X]-2\operatorname {Cov} [X,Y]+\operatorname {Var} [Y].}
 
Particularly, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are independent from each other, then: 
 
 
 
 Var
 ⁡
 [
 X
 −
 Y
 ]
 =
 Var
 ⁡
 [
 X
 ]
 +
 Var
 ⁡
 [
 Y
 ]
 .
 
 
 {\displaystyle \operatorname {Var} [X-Y]=\operatorname {Var} [X]+\operatorname {Var} [Y].}
 
 That is, for <a href="/facts/Independent_random_variables/NUzQtnUL">independent random variables</a> the variance is the same for additions and subtractions: 
 
 
 
 Var
 ⁡
 [
 X
 +
 Y
 ]
 =
 Var
 ⁡
 [
 X
 −
 Y
 ]
 =
 Var
 ⁡
 [
 Y
 −
 X
 ]
 =
 Var
 ⁡
 [
 −
 X
 −
 Y
 ]
 .
 
 
 {\displaystyle \operatorname {Var} [X+Y]=\operatorname {Var} [X-Y]=\operatorname {Var} [Y-X]=\operatorname {Var} [-X-Y].}
 
</li>
<li><a href="/facts/Multiplication/VtcUjpNv">Multiplication</a>: 
 
 
 
 Var
 ⁡
 [
 Z
 ]
 =
 Var
 ⁡
 [
 X
 Y
 ]
 =
 Var
 ⁡
 [
 Y
 X
 ]
 .
 
 
 {\displaystyle \operatorname {Var} [Z]=\operatorname {Var} [XY]=\operatorname {Var} [YX].}
 
 Particularly, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are independent from each other, then: 
 
 
 
 
 
 
 
 Var
 ⁡
 [
 X
 Y
 ]
 
 
 
 =
 E
 ⁡
 [
 
 X
 
 2
 
 
 ]
 ⋅
 E
 ⁡
 [
 
 Y
 
 2
 
 
 ]
 −
 
 
 
 (
 
 E
 ⁡
 [
 X
 ]
 ⋅
 E
 ⁡
 [
 Y
 ]
 
 )
 
 
 
 2
 
 
 
 
 
 
 
 
 =
 Var
 ⁡
 [
 X
 ]
 ⋅
 Var
 ⁡
 [
 Y
 ]
 +
 Var
 ⁡
 [
 X
 ]
 ⋅
 
 
 
 (
 
 E
 ⁡
 [
 Y
 ]
 
 )
 
 
 
 2
 
 
 +
 Var
 ⁡
 [
 Y
 ]
 ⋅
 
 
 
 (
 
 E
 ⁡
 [
 X
 ]
 
 )
 
 
 
 2
 
 
 .
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}\operatorname {Var} [XY]&=\operatorname {E} [X^{2}]\cdot \operatorname {E} [Y^{2}]-{\left(\operatorname {E} [X]\cdot \operatorname {E} [Y]\right)}^{2}\\[2pt]&=\operatorname {Var} [X]\cdot \operatorname {Var} [Y]+\operatorname {Var} [X]\cdot {\left(\operatorname {E} [Y]\right)}^{2}+\operatorname {Var} [Y]\cdot {\left(\operatorname {E} [X]\right)}^{2}.\end{aligned}}}
 
</li>
<li><a href="/facts/Division_(mathematics)/FqLKNkmq">Division</a>: 
 
 
 
 Var
 ⁡
 [
 Z
 ]
 =
 Var
 ⁡
 [
 X
 
 /
 
 Y
 ]
 =
 Var
 ⁡
 [
 X
 ⋅
 (
 1
 
 /
 
 Y
 )
 ]
 =
 Var
 ⁡
 [
 (
 1
 
 /
 
 Y
 )
 ⋅
 X
 ]
 .
 
 
 {\displaystyle \operatorname {Var} [Z]=\operatorname {Var} [X/Y]=\operatorname {Var} [X\cdot (1/Y)]=\operatorname {Var} [(1/Y)\cdot X].}
 
 Particularly, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are independent from each other, then: 
 
 
 
 
 
 
 
 Var
 ⁡
 [
 X
 
 /
 
 Y
 ]
 
 
 
 =
 E
 ⁡
 [
 
 X
 
 2
 
 
 ]
 ⋅
 E
 ⁡
 [
 1
 
 /
 
 
 Y
 
 2
 
 
 ]
 −
 
 
 
 (
 
 E
 ⁡
 [
 X
 ]
 ⋅
 E
 ⁡
 [
 1
 
 /
 
 Y
 ]
 
 )
 
 
 
 2
 
 
 
 
 
 
 
 
 =
 Var
 ⁡
 [
 X
 ]
 ⋅
 Var
 ⁡
 [
 1
 
 /
 
 Y
 ]
 +
 Var
 ⁡
 [
 X
 ]
 ⋅
 
 
 
 (
 
 E
 ⁡
 [
 1
 
 /
 
 Y
 ]
 
 )
 
 
 
 2
 
 
 +
 Var
 ⁡
 [
 1
 
 /
 
 Y
 ]
 ⋅
 
 
 
 (
 
 E
 ⁡
 [
 X
 ]
 
 )
 
 
 
 2
 
 
 .
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}\operatorname {Var} [X/Y]&=\operatorname {E} [X^{2}]\cdot \operatorname {E} [1/Y^{2}]-{\left(\operatorname {E} [X]\cdot \operatorname {E} [1/Y]\right)}^{2}\\[2pt]&=\operatorname {Var} [X]\cdot \operatorname {Var} [1/Y]+\operatorname {Var} [X]\cdot {\left(\operatorname {E} [1/Y]\right)}^{2}+\operatorname {Var} [1/Y]\cdot {\left(\operatorname {E} [X]\right)}^{2}.\end{aligned}}}
 
</li>
<li><a href="/facts/Exponentiation/pac6QY2g">Exponentiation</a>: 
 
 
 
 Var
 ⁡
 [
 Z
 ]
 =
 Var
 ⁡
 [
 
 X
 
 Y
 
 
 ]
 =
 Var
 ⁡
 [
 
 e
 
 Y
 ln
 ⁡
 (
 X
 )
 
 
 ]
 
 
 {\displaystyle \operatorname {Var} [Z]=\operatorname {Var} [X^{Y}]=\operatorname {Var} [e^{Y\ln(X)}]}
 
</li></ul>
where 
 
 
 
 Cov
 ⁡
 [
 X
 ,
 Y
 ]
 =
 Cov
 ⁡
 [
 Y
 ,
 X
 ]
 
 
 {\displaystyle \operatorname {Cov} [X,Y]=\operatorname {Cov} [Y,X]}
 
 represents the covariance operator between random variables 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
.
The variance of a random variable can also be expressed directly in terms of the covariance or in terms of the expected value:

 
 
 
 Var
 ⁡
 [
 X
 ]
 =
 Cov
 ⁡
 (
 X
 ,
 X
 )
 =
 E
 ⁡
 [
 
 X
 
 2
 
 
 ]
 −
 E
 ⁡
 [
 X
 
 ]
 
 2
 
 
 
 
 {\displaystyle \operatorname {Var} [X]=\operatorname {Cov} (X,X)=\operatorname {E} [X^{2}]-\operatorname {E} [X]^{2}}

If any of the random variables is replaced by a deterministic variable or by a constant value (
 
 
 
 k
 
 
 {\displaystyle k}
 
), the previous properties remain valid considering that 
 
 
 
 Pr
 (
 X
 =
 k
 )
 =
 1
 
 
 {\displaystyle \Pr(X=k)=1}
 
 and 
 
 
 
 E
 ⁡
 [
 X
 ]
 =
 k
 
 
 {\displaystyle \operatorname {E} [X]=k}
 
, 
 
 
 
 Var
 ⁡
 [
 X
 ]
 =
 0
 
 
 {\displaystyle \operatorname {Var} [X]=0}
 
 and 
 
 
 
 Cov
 ⁡
 [
 Y
 ,
 k
 ]
 =
 0
 
 
 {\displaystyle \operatorname {Cov} [Y,k]=0}
 
. Special cases are the addition and multiplication of a random variable with a deterministic variable or a constant, where:

<ul><li>
 
 
 
 Var
 ⁡
 [
 k
 +
 Y
 ]
 =
 Var
 ⁡
 [
 Y
 ]
 
 
 {\displaystyle \operatorname {Var} [k+Y]=\operatorname {Var} [Y]}
 
</li>
<li>
 
 
 
 Var
 ⁡
 [
 k
 Y
 ]
 =
 
 k
 
 2
 
 
 Var
 ⁡
 [
 Y
 ]
 
 
 {\displaystyle \operatorname {Var} [kY]=k^{2}\operatorname {Var} [Y]}
 
</li></ul>
If 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 is defined as a general non-linear algebraic function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 of a random variable 
 
 
 
 X
 
 
 {\displaystyle X}
 
, then:

 
 
 
 Var
 ⁡
 [
 Z
 ]
 =
 Var
 ⁡
 [
 f
 (
 X
 )
 ]
 ≠
 f
 (
 Var
 ⁡
 [
 X
 ]
 )
 
 
 {\displaystyle \operatorname {Var} [Z]=\operatorname {Var} [f(X)]\neq f(\operatorname {Var} [X])}

The exact value of the variance of the non-linear function will depend on the particular probability distribution of the random variable 
 
 
 
 X
 
 
 {\displaystyle X}
 
.

<h2 id="covariance-algebra-for-random-variables">Covariance algebra for random variables</h2>
The covariance (
 
 
 
 Cov
 ⁡
 [
 Z
 ,
 X
 ]
 
 
 {\displaystyle \operatorname {Cov} [Z,X]}
 
) between the random variable 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 resulting from an algebraic operation and the random variable 
 
 
 
 X
 
 
 {\displaystyle X}
 
 can be calculated using the following set of rules:

<ul><li><a href="/facts/Addition/apFWJBFB">Addition</a>: 
 
 
 
 Cov
 ⁡
 [
 Z
 ,
 X
 ]
 =
 Cov
 ⁡
 [
 X
 +
 Y
 ,
 X
 ]
 =
 Var
 ⁡
 [
 X
 ]
 +
 Cov
 ⁡
 [
 X
 ,
 Y
 ]
 .
 
 
 {\displaystyle \operatorname {Cov} [Z,X]=\operatorname {Cov} [X+Y,X]=\operatorname {Var} [X]+\operatorname {Cov} [X,Y].}
 
 If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are <a href="/facts/Independence_(probability_theory)/NUzQtnUL">independent</a> from each other, then: 
 
 
 
 Cov
 ⁡
 [
 X
 +
 Y
 ,
 X
 ]
 =
 Var
 ⁡
 [
 X
 ]
 .
 
 
 {\displaystyle \operatorname {Cov} [X+Y,X]=\operatorname {Var} [X].}
 
</li>
<li><a href="/facts/Subtraction/dEUrp6k0">Subtraction</a>: 
 
 
 
 Cov
 ⁡
 [
 Z
 ,
 X
 ]
 =
 Cov
 ⁡
 [
 X
 −
 Y
 ,
 X
 ]
 =
 Var
 ⁡
 [
 X
 ]
 −
 Cov
 ⁡
 [
 X
 ,
 Y
 ]
 .
 
 
 {\displaystyle \operatorname {Cov} [Z,X]=\operatorname {Cov} [X-Y,X]=\operatorname {Var} [X]-\operatorname {Cov} [X,Y].}
 
 If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are independent from each other, then: 
 
 
 
 Cov
 ⁡
 [
 X
 −
 Y
 ,
 X
 ]
 =
 Var
 ⁡
 [
 X
 ]
 .
 
 
 {\displaystyle \operatorname {Cov} [X-Y,X]=\operatorname {Var} [X].}
 
</li>
<li><a href="/facts/Multiplication/VtcUjpNv">Multiplication</a>: 
 
 
 
 Cov
 ⁡
 [
 Z
 ,
 X
 ]
 =
 Cov
 ⁡
 [
 X
 Y
 ,
 X
 ]
 =
 E
 ⁡
 [
 
 X
 
 2
 
 
 Y
 ]
 −
 E
 ⁡
 [
 X
 Y
 ]
 E
 ⁡
 [
 X
 ]
 .
 
 
 {\displaystyle \operatorname {Cov} [Z,X]=\operatorname {Cov} [XY,X]=\operatorname {E} [X^{2}Y]-\operatorname {E} [XY]\operatorname {E} [X].}
 
 If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are independent from each other, then: 
 
 
 
 Cov
 ⁡
 [
 X
 Y
 ,
 X
 ]
 =
 Var
 ⁡
 [
 X
 ]
 ⋅
 E
 ⁡
 [
 Y
 ]
 .
 
 
 {\displaystyle \operatorname {Cov} [XY,X]=\operatorname {Var} [X]\cdot \operatorname {E} [Y].}
 
</li>
<li><a href="/facts/Division_(mathematics)/FqLKNkmq">Division</a> (covariance with respect to the numerator): 
 
 
 
 Cov
 ⁡
 [
 Z
 ,
 X
 ]
 =
 Cov
 ⁡
 [
 X
 
 /
 
 Y
 ,
 X
 ]
 =
 E
 ⁡
 [
 
 X
 
 2
 
 
 
 /
 
 Y
 ]
 −
 E
 ⁡
 [
 X
 
 /
 
 Y
 ]
 E
 ⁡
 [
 X
 ]
 .
 
 
 {\displaystyle \operatorname {Cov} [Z,X]=\operatorname {Cov} [X/Y,X]=\operatorname {E} [X^{2}/Y]-\operatorname {E} [X/Y]\operatorname {E} [X].}
 
 If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are independent from each other, then: 
 
 
 
 Cov
 ⁡
 [
 X
 
 /
 
 Y
 ,
 X
 ]
 =
 Var
 ⁡
 [
 X
 ]
 ⋅
 E
 ⁡
 [
 1
 
 /
 
 Y
 ]
 .
 
 
 {\displaystyle \operatorname {Cov} [X/Y,X]=\operatorname {Var} [X]\cdot \operatorname {E} [1/Y].}
 
</li>
<li><a href="/facts/Division_(mathematics)/FqLKNkmq">Division</a> (covariance with respect to the denominator): 
 
 
 
 Cov
 ⁡
 [
 Z
 ,
 X
 ]
 =
 Cov
 ⁡
 [
 Y
 
 /
 
 X
 ,
 X
 ]
 =
 E
 ⁡
 [
 Y
 ]
 −
 E
 ⁡
 [
 Y
 
 /
 
 X
 ]
 E
 ⁡
 [
 X
 ]
 .
 
 
 {\displaystyle \operatorname {Cov} [Z,X]=\operatorname {Cov} [Y/X,X]=\operatorname {E} [Y]-\operatorname {E} [Y/X]\operatorname {E} [X].}
 
 If 
 
 
 
 X
 
 
 {\displaystyle X}
 
 and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are independent from each other, then: 
 
 
 
 Cov
 ⁡
 [
 Y
 
 /
 
 X
 ,
 X
 ]
 =
 E
 ⁡
 [
 Y
 ]
 ⋅
 (
 1
 −
 E
 ⁡
 [
 X
 ]
 ⋅
 E
 ⁡
 [
 1
 
 /
 
 X
 ]
 )
 .
 
 
 {\displaystyle \operatorname {Cov} [Y/X,X]=\operatorname {E} [Y]\cdot (1-\operatorname {E} [X]\cdot \operatorname {E} [1/X]).}
 
</li>
<li><a href="/facts/Exponentiation/pac6QY2g">Exponentiation</a> (covariance with respect to the base): 
 
 
 
 Cov
 ⁡
 [
 Z
 ,
 X
 ]
 =
 Cov
 ⁡
 [
 
 X
 
 Y
 
 
 ,
 X
 ]
 =
 E
 ⁡
 [
 
 X
 
 Y
 +
 1
 
 
 ]
 −
 E
 ⁡
 [
 
 X
 
 Y
 
 
 ]
 E
 ⁡
 [
 X
 ]
 .
 
 
 {\displaystyle \operatorname {Cov} [Z,X]=\operatorname {Cov} [X^{Y},X]=\operatorname {E} [X^{Y+1}]-\operatorname {E} [X^{Y}]\operatorname {E} [X].}
 
</li>
<li><a href="/facts/Exponentiation/pac6QY2g">Exponentiation</a> (covariance with respect to the power): 
 
 
 
 Cov
 ⁡
 [
 Z
 ,
 X
 ]
 =
 Cov
 ⁡
 [
 
 Y
 
 X
 
 
 ,
 X
 ]
 =
 E
 ⁡
 [
 X
 
 Y
 
 X
 
 
 ]
 −
 E
 ⁡
 [
 
 Y
 
 X
 
 
 ]
 E
 ⁡
 [
 X
 ]
 .
 
 
 {\displaystyle \operatorname {Cov} [Z,X]=\operatorname {Cov} [Y^{X},X]=\operatorname {E} [XY^{X}]-\operatorname {E} [Y^{X}]\operatorname {E} [X].}
 
</li></ul>
The covariance of a random variable can also be expressed directly in terms of the expected value:

 
 
 
 Cov
 ⁡
 (
 X
 ,
 Y
 )
 =
 E
 ⁡
 [
 X
 Y
 ]
 −
 E
 ⁡
 [
 X
 ]
 E
 ⁡
 [
 Y
 ]
 
 
 {\displaystyle \operatorname {Cov} (X,Y)=\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y]}

If any of the random variables is replaced by a deterministic variable or by a constant value (
 
 
 
 k
 
 
 {\displaystyle k}
 
), the previous properties remain valid considering that 
 
 
 
 E
 ⁡
 [
 k
 ]
 =
 k
 
 
 {\displaystyle \operatorname {E} [k]=k}
 
, 
 
 
 
 Var
 ⁡
 [
 k
 ]
 =
 0
 
 
 {\displaystyle \operatorname {Var} [k]=0}
 
 and 
 
 
 
 Cov
 ⁡
 [
 X
 ,
 k
 ]
 =
 0
 
 
 {\displaystyle \operatorname {Cov} [X,k]=0}
 
.
If 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 is defined as a general non-linear algebraic function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 of a random variable 
 
 
 
 X
 
 
 {\displaystyle X}
 
, then:

 
 
 
 Cov
 ⁡
 [
 Z
 ,
 X
 ]
 =
 Cov
 ⁡
 [
 f
 (
 X
 )
 ,
 X
 ]
 =
 E
 ⁡
 [
 X
 f
 (
 X
 )
 ]
 −
 E
 ⁡
 [
 f
 (
 X
 )
 ]
 E
 ⁡
 [
 X
 ]
 
 
 {\displaystyle \operatorname {Cov} [Z,X]=\operatorname {Cov} [f(X),X]=\operatorname {E} [Xf(X)]-\operatorname {E} [f(X)]\operatorname {E} [X]}

The exact value of the covariance of the non-linear function will depend on the particular probability distribution of the random variable 
 
 
 
 X
 
 
 {\displaystyle X}
 
.

<h2 id="approximations-by-taylor-series-expansions-of-moments">Approximations by Taylor series expansions of moments</h2>
If the <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moments</a> of a certain random variable 
 
 
 
 X
 
 
 {\displaystyle X}
 
 are known (or can be determined by integration if the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> is known), then it is possible to approximate the expected value of any general non-linear function 
 
 
 
 f
 (
 X
 )
 
 
 {\displaystyle f(X)}
 
 as a <a href="/facts/Taylor_expansions_for_the_moments_of_functions_of_random_variables/AgoGBcbS">Taylor series expansion of the moments</a>, as follows:

 
 
 
 f
 (
 X
 )
 =
 
 ∑
 
 n
 =
 0
 
 
 ∞
 
 
 
 
 1
 
 n
 !
 
 
 
 
 
 (
 
 
 
 
 d
 
 n
 
 
 f
 
 
 d
 
 X
 
 n
 
 
 
 
 
 )
 
 
 X
 =
 μ
 
 
 
 
 
 (
 
 X
 −
 μ
 
 )
 
 
 
 n
 
 
 ,
 
 
 {\displaystyle f(X)=\sum _{n=0}^{\infty }{\frac {1}{n!}}\left({\frac {d^{n}f}{dX^{n}}}\right)_{X=\mu }{\left(X-\mu \right)}^{n},}

where 
 
 
 
 μ
 =
 E
 ⁡
 [
 X
 ]
 
 
 {\displaystyle \mu =\operatorname {E} [X]}
 
 is the mean value of 
 
 
 
 X
 
 
 {\displaystyle X}
 
.

 
 
 
 
 
 
 
 E
 ⁡
 [
 f
 (
 X
 )
 ]
 
 
 
 =
 E
 ⁡
 
 [
 
 
 ∑
 
 n
 =
 0
 
 
 ∞
 
 
 
 
 1
 
 n
 !
 
 
 
 
 
 (
 
 
 
 
 d
 
 n
 
 
 f
 
 
 d
 
 X
 
 n
 
 
 
 
 
 )
 
 
 X
 =
 μ
 
 
 
 
 
 (
 
 X
 −
 μ
 
 )
 
 
 
 n
 
 
 
 ]
 
 
 
 
 
 
 
 =
 
 ∑
 
 n
 =
 0
 
 
 ∞
 
 
 
 
 1
 
 n
 !
 
 
 
 
 
 (
 
 
 
 
 d
 
 n
 
 
 f
 
 
 d
 
 X
 
 n
 
 
 
 
 
 )
 
 
 X
 =
 μ
 
 
 E
 ⁡
 
 [
 
 
 
 (
 
 X
 −
 μ
 
 )
 
 
 
 n
 
 
 ]
 
 
 
 
 
 
 
 =
 
 ∑
 
 n
 =
 0
 
 
 ∞
 
 
 
 
 1
 
 n
 !
 
 
 
 
 
 (
 
 
 
 
 d
 
 n
 
 
 f
 
 
 d
 
 X
 
 n
 
 
 
 
 
 )
 
 
 X
 =
 μ
 
 
 
 μ
 
 n
 
 
 (
 X
 )
 ,
 
 
 
 
 
 
 {\displaystyle {\begin{aligned}\operatorname {E} [f(X)]&=\operatorname {E} \left[\sum _{n=0}^{\infty }{\frac {1}{n!}}\left({d^{n}f \over dX^{n}}\right)_{X=\mu }{\left(X-\mu \right)}^{n}\right]\\&=\sum _{n=0}^{\infty }{\frac {1}{n!}}\left({\frac {d^{n}f}{dX^{n}}}\right)_{X=\mu }\operatorname {E} \left[{\left(X-\mu \right)}^{n}\right]\\&=\sum _{n=0}^{\infty }{\frac {1}{n!}}\left({d^{n}f \over dX^{n}}\right)_{X=\mu }\mu _{n}(X),\end{aligned}}}

where 
 
 
 
 
 μ
 
 n
 
 
 (
 X
 )
 =
 E
 ⁡
 [
 (
 X
 −
 μ
 
 )
 
 n
 
 
 ]
 
 
 {\displaystyle \mu _{n}(X)=\operatorname {E} [(X-\mu )^{n}]}
 
 is the n-th moment of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 about its mean. Note that by their definition, 
 
 
 
 
 μ
 
 0
 
 
 (
 X
 )
 =
 1
 
 
 {\displaystyle \mu _{0}(X)=1}
 
 and 
 
 
 
 
 μ
 
 1
 
 
 (
 X
 )
 =
 0
 
 
 {\displaystyle \mu _{1}(X)=0}
 
. The first order term always vanishes but was kept to obtain a closed form expression.
Then,

 
 
 
 E
 ⁡
 [
 f
 (
 X
 )
 ]
 ≈
 
 ∑
 
 n
 =
 0
 
 
 
 n
 
 max
 
 
 
 
 
 
 1
 
 n
 !
 
 
 
 
 
 (
 
 
 
 
 d
 
 n
 
 
 f
 
 
 d
 
 X
 
 n
 
 
 
 
 
 )
 
 
 X
 =
 μ
 
 
 
 μ
 
 n
 
 
 (
 X
 )
 ,
 
 
 {\displaystyle \operatorname {E} [f(X)]\approx \sum _{n=0}^{n_{\max }}{\frac {1}{n!}}\left({\frac {d^{n}f}{dX^{n}}}\right)_{X=\mu }\mu _{n}(X),}

where the Taylor expansion is truncated after the 
 
 
 
 
 n
 
 max
 
 
 
 
 {\displaystyle n_{\max }}
 
-th moment.
Particularly for functions of <a href="/facts/Normal_random_variable/UapjjPyQ">normal random variables</a>, it is possible to obtain a Taylor expansion in terms of the <a href="/facts/Standard_normal_distribution/UapjjPyQ">standard normal distribution</a>:<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

 
 
 
 f
 (
 X
 )
 =
 
 ∑
 
 n
 =
 0
 
 
 ∞
 
 
 
 
 
 σ
 
 n
 
 
 
 n
 !
 
 
 
 
 
 (
 
 
 
 
 d
 
 n
 
 
 f
 
 
 d
 
 X
 
 n
 
 
 
 
 
 )
 
 
 X
 =
 μ
 
 
 
 μ
 
 n
 
 
 (
 Z
 )
 ,
 
 
 {\displaystyle f(X)=\sum _{n=0}^{\infty }{\frac {\sigma ^{n}}{n!}}\left({\frac {d^{n}f}{dX^{n}}}\right)_{X=\mu }\mu _{n}(Z),}
 
where 
 
 
 
 X
 ∼
 N
 (
 μ
 ,
 
 σ
 
 2
 
 
 )
 
 
 {\displaystyle X\sim N(\mu ,\sigma ^{2})}
 
 is a normal random variable, and 
 
 
 
 Z
 ∼
 N
 (
 0
 ,
 1
 )
 
 
 {\displaystyle Z\sim N(0,1)}
 
 is the standard normal distribution. Thus,

 
 
 
 E
 ⁡
 [
 f
 (
 X
 )
 ]
 ≈
 
 ∑
 
 n
 =
 0
 
 
 
 n
 
 max
 
 
 
 
 
 
 
 σ
 
 n
 
 
 
 n
 !
 
 
 
 
 
 (
 
 
 
 
 d
 
 n
 
 
 f
 
 
 d
 
 X
 
 n
 
 
 
 
 
 )
 
 
 X
 =
 μ
 
 
 
 μ
 
 n
 
 
 (
 Z
 )
 ,
 
 
 {\displaystyle \operatorname {E} [f(X)]\approx \sum _{n=0}^{n_{\max }}{\sigma ^{n} \over n!}\left({d^{n}f \over dX^{n}}\right)_{X=\mu }\mu _{n}(Z),}
 
 where the moments of the standard normal distribution are given by:

 
 
 
 
 μ
 
 n
 
 
 (
 Z
 )
 =
 
 
 {
 
 
 
 
 ∏
 
 i
 =
 1
 
 
 n
 
 /
 
 2
 
 
 (
 2
 i
 −
 1
 )
 ,
 
 
 
 if 
 
 n
 
  is even
 
 
 
 
 
 0
 ,
 
 
 
 if 
 
 n
 
  is odd
 
 
 
 
 
 
 
 
 
 {\displaystyle \mu _{n}(Z)={\begin{cases}\prod _{i=1}^{n/2}(2i-1),&{\text{if }}n{\text{ is even}}\\0,&{\text{if }}n{\text{ is odd}}\end{cases}}}

Similarly for normal random variables, it is also possible to approximate the variance of the non-linear function as a Taylor series expansion as:

 
 
 
 Var
 ⁡
 [
 f
 (
 X
 )
 ]
 ≈
 
 ∑
 
 n
 =
 1
 
 
 
 n
 
 max
 
 
 
 
 
 
 (
 
 
 
 
 σ
 
 n
 
 
 
 n
 !
 
 
 
 
 
 (
 
 
 
 
 d
 
 n
 
 
 f
 
 
 d
 
 X
 
 n
 
 
 
 
 
 )
 
 
 X
 =
 μ
 
 
 
 )
 
 
 2
 
 
 Var
 ⁡
 [
 
 Z
 
 n
 
 
 ]
 +
 
 ∑
 
 n
 =
 1
 
 
 
 n
 
 max
 
 
 
 
 
 ∑
 
 m
 ≠
 n
 
 
 
 
 
 σ
 
 n
 +
 m
 
 
 
 n
 !
 m
 !
 
 
 
 
 
 (
 
 
 
 
 d
 
 n
 
 
 f
 
 
 d
 
 X
 
 n
 
 
 
 
 
 )
 
 
 X
 =
 μ
 
 
 
 
 (
 
 
 
 
 d
 
 m
 
 
 f
 
 
 d
 
 X
 
 m
 
 
 
 
 
 )
 
 
 X
 =
 μ
 
 
 Cov
 ⁡
 [
 
 Z
 
 n
 
 
 ,
 
 Z
 
 m
 
 
 ]
 ,
 
 
 {\displaystyle \operatorname {Var} [f(X)]\approx \sum _{n=1}^{n_{\max }}\left({\sigma ^{n} \over n!}\left({d^{n}f \over dX^{n}}\right)_{X=\mu }\right)^{2}\operatorname {Var} [Z^{n}]+\sum _{n=1}^{n_{\max }}\sum _{m\neq n}{\frac {\sigma ^{n+m}}{n!m!}}\left({d^{n}f \over dX^{n}}\right)_{X=\mu }\left({d^{m}f \over dX^{m}}\right)_{X=\mu }\operatorname {Cov} [Z^{n},Z^{m}],}

where

Var
        ⁡
        [
        
          Z
          
            n
          
        
        ]
        =
        
          
            {
            
              
                
                  
                    ∏
                    
                      i
                      =
                      1
                    
                    
                      n
                    
                  
                  (
                  2
                  i
                  −
                  1
                  )
                  −
                  
                    ∏
                    
                      i
                      =
                      1
                    
                    
                      n
                      
                        /
                      
                      2
                    
                  
                  (
                  2
                  i
                  −
                  1
                  
                    )
                    
                      2
                    
                  
                  ,
                
                
                  
                    if 
                  
                  n
                  
                     is even
                  
                
              
              
                
                  
                    ∏
                    
                      i
                      =
                      1
                    
                    
                      n
                    
                  
                  (
                  2
                  i
                  −
                  1
                  )
                  ,
                
                
                  
                    if 
                  
                  n
                  
                     is odd
                  
                  ,
                
              
            
            
          
        
      
    
    {\displaystyle \operatorname {Var} [Z^{n}]={\begin{cases}\prod _{i=1}^{n}(2i-1)-\prod _{i=1}^{n/2}(2i-1)^{2},&{\text{if }}n{\text{ is even}}\\\prod _{i=1}^{n}(2i-1),&{\text{if }}n{\text{ is odd}},\end{cases}}}

and

Cov
        ⁡
        [
        
          Z
          
            n
          
        
        ,
        
          Z
          
            m
          
        
        ]
        =
        
          
            {
            
              
                
                  
                    ∏
                    
                      i
                      =
                      1
                    
                    
                      (
                      n
                      +
                      m
                      )
                      
                        /
                      
                      2
                    
                  
                  (
                  2
                  i
                  −
                  1
                  )
                  −
                  
                    ∏
                    
                      i
                      =
                      1
                    
                    
                      n
                      
                        /
                      
                      2
                    
                  
                  (
                  2
                  i
                  −
                  1
                  )
                  
                    ∏
                    
                      j
                      =
                      1
                    
                    
                      m
                      
                        /
                      
                      2
                    
                  
                  (
                  2
                  j
                  −
                  1
                  )
                  ,
                
                
                  
                    if 
                  
                  n
                  
                     and 
                  
                  m
                  
                     are even
                  
                
              
              
                
                  
                    ∏
                    
                      i
                      =
                      1
                    
                    
                      (
                      n
                      +
                      m
                      )
                      
                        /
                      
                      2
                    
                  
                  (
                  2
                  i
                  −
                  1
                  )
                  ,
                
                
                  
                    if 
                  
                  n
                  
                     and 
                  
                  m
                  
                     are odd
                  
                
              
              
                
                  0
                  ,
                
                
                  
                    otherwise
                  
                
              
            
            
          
        
      
    
    {\displaystyle \operatorname {Cov} [Z^{n},Z^{m}]={\begin{cases}\prod _{i=1}^{(n+m)/2}(2i-1)-\prod _{i=1}^{n/2}(2i-1)\prod _{j=1}^{m/2}(2j-1),&{\text{if }}n{\text{ and }}m{\text{ are even}}\\\prod _{i=1}^{(n+m)/2}(2i-1),&{\text{if }}n{\text{ and }}m{\text{ are odd}}\\0,&{\text{otherwise}}\end{cases}}}

<h2 id="algebra-of-complex-random-variables">Algebra of complex random variables</h2>
In the <a href="/facts/Algebra/nMdqczjo">algebraic</a> <a href="/facts/Axiom/mLjP14Mc">axiomatization</a> of <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, the primary concept is not that of probability of an event, but rather that of a <a href="/facts/Random_variable/TwTBXnLT">random variable</a>. <a href="/facts/Probability_distribution/EpsKKVRu">Probability distributions</a> are determined by assigning an <a href="/facts/Expected_value/1XV0JKL8">expectation</a> to each random variable. The <a href="/facts/Measure_(mathematics)/yCq7zlI4">measurable space</a> and the probability measure arise from the random variables and expectations by means of well-known <a href="/facts/Representation_theorem/Fu6DYyGD">representation theorems</a> 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:

<ol><li><a href="/facts/Complex_number/2w7mqI7e">complex</a> constants are possible <a href="/facts/Realization_(probability)/lDklUzqX">realizations</a> of a random variable;</li>
<li>the sum of two random variables is a random variable;</li>
<li>the product of two random variables is a random variable;</li>
<li>addition and multiplication of random variables are both <a href="/facts/Commutative/WaYxJLxd">commutative</a>; and</li>
<li>there is a notion of conjugation of random variables, satisfying (XY)* = Y*X* and X** = X for all random variables X,Y and coinciding with complex conjugation if X is a constant.</li></ol>
This means that random variables form complex commutative <a href="/facts/*-algebra/vcE5TWnT">*-algebras</a>. If X = X* then the random variable X is called "real".
An expectation E on an algebra A of random variables is a normalized, positive <a href="/facts/Linear_functional/oGh7Asrz">linear functional</a>. What this means is that

<ol><li>E[k] = k where k is a constant;</li>
<li>E[X*X] ≥ 0 for all random variables X;</li>
<li>E[X + Y] = E[X] + E[Y] for all random variables X and Y; and</li>
<li>E[kX] = kE[X] if k is a constant.</li></ol>
One may generalize this setup, allowing the algebra to be noncommutative. This leads to other areas of noncommutative probability such as <a href="/facts/Quantum_probability/XAsF5Ndf">quantum probability</a>, <a href="/facts/Random_matrix_theory/Yqo1lwFW">random matrix theory</a>, and <a href="/facts/Free_probability/xV0lMQry">free probability</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Relationships_among_probability_distributions/2aS20E3h">Relationships among probability distributions</a></li>
<li><a href="/facts/Ratio_distribution/5rUEmWkG">Ratio distribution</a>
<ul><li><a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy distribution</a></li>
<li><a href="/facts/Slash_distribution/NxJ6murS">Slash distribution</a></li></ul></li>
<li><a href="/facts/Inverse_distribution/VVvNkMXu">Inverse distribution</a></li>
<li><a href="/facts/Product_distribution/RVIMz0xc">Product distribution</a></li>
<li><a href="/facts/Mellin_transform/M0087pMe">Mellin transform</a></li>
<li><a href="/facts/Sum_of_normally_distributed_random_variables/IFKlcaLC">Sum of normally distributed random variables</a></li>
<li><a href="/facts/List_of_convolutions_of_probability_distributions/d2MBrphK">List of convolutions of probability distributions</a> – the <a href="/facts/Probability_measure/8BjhgoPR">probability measure</a> of the sum of <a href="/facts/Independent_random_variables/NUzQtnUL">independent random variables</a> is the <a href="/facts/Convolution/PVPrdz9J">convolution</a> of their probability measures.</li>
<li><a href="/facts/Law_of_total_expectation/zSqn8uVj">Law of total expectation</a></li>
<li><a href="/facts/Law_of_total_variance/hDEf2nqK">Law of total variance</a></li>
<li><a href="/facts/Law_of_total_covariance/VvlxJrM0">Law of total covariance</a></li>
<li><a href="/facts/Law_of_total_cumulance/cfna8F0s">Law of total cumulance</a></li>
<li><a href="/facts/Taylor_expansions_for_the_moments_of_functions_of_random_variables/AgoGBcbS">Taylor expansions for the moments of functions of random variables</a></li>
<li><a href="/facts/Delta_method/C2djfrlx">Delta method</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Peter_Whittle_(mathematician)/eGoN2BdP">Whittle, Peter</a> (2000). <a href="https://www.springer.com/statistics/book/978-0-387-98955-6">Probability via Expectation</a> (4th ed.). New York, NY: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-98955-6. Retrieved 24 September 2012.</li>
<li>Springer, Melvin Dale (1979). <a href="https://books.google.com/books?id=qUDvAAAAMAAJ">The Algebra of Random Variables</a>. <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">Wiley</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-01406-0. Retrieved 24 September 2012.</li>
<li><a href="https://www.encyclopediaofmath.org/index.php?title=Measure_algebra">"Measure algebra"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Hernandez, Hugo (2016). "Modelling the effect of fluctuation in nonlinear systems using variance algebra - Application to light scattering of ideal gases". ForsChem Research Reports. 2016–1. doi:10.13140/rg.2.2.36501.52969. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Algebra of random variables open-in-new

Algebra of random variables