Editing Independence (probability theory) (section)

==Properties==
===Self-independence===
Note that an event is independent of itself if and only if

:<math>\mathrm{P}(A) = \mathrm{P}(A \cap A) = \mathrm{P}(A) \cdot \mathrm{P}(A) \iff \mathrm{P}(A) = 0 \text{ or } \mathrm{P}(A) = 1.</math>

Thus an event is independent of itself if and only if it [[almost surely]] occurs or its [[Complement (set theory)|complement]] almost surely occurs; this fact is useful when proving [[zero–one law]]s.<ref>{{cite book|last=Durrett|first=Richard|author-link=Rick Durrett|title=Probability: theory and examples|edition=Second|year=1996}} page 62</ref>

===Expectation and covariance===
{{main|Correlation and dependence}}
If <math>X</math> and <math>Y</math> are statistically independent random variables, then the [[expected value|expectation operator]] <math>\operatorname{E}</math> has the property

:<math>\operatorname{E}[X^n Y^m] = \operatorname{E}[X^n] \operatorname{E}[Y^m],</math><ref name=JakemanBook>{{cite book | author=E Jakeman| title=MODELING FLUCTUATIONS IN SCATTERED WAVES| isbn=978-0-7503-1005-5}}</ref>{{rp|p. 10}}

and the [[covariance]] <math>\operatorname{cov}[X,Y]</math> is zero, as follows from

:<math>\operatorname{cov}[X,Y] = \operatorname{E}[X Y] - \operatorname{E}[X] \operatorname{E}[Y].</math>

The converse does not hold: if two random variables have a covariance of 0 they still may be not independent. {{See also|Uncorrelatedness (probability theory)}}

Similarly for two stochastic processes <math>\left\{ X_t \right\}_{t\in\mathcal{T}}</math> and <math>\left\{ Y_t \right\}_{t\in\mathcal{T}}</math>: If they are independent, then they are [[Uncorrelatedness (probability theory)|uncorrelated]].<ref name=KunIlPark>{{cite book | author=Park, Kun Il| title=Fundamentals of Probability and Stochastic Processes with Applications to Communications| publisher=Springer | year=2018 | isbn=978-3-319-68074-3}}</ref>{{rp|p. 151}}

===Characteristic function===
Two random variables <math>X</math> and <math>Y</math> are independent if and only if the [[characteristic function (probability theory)|characteristic function]] of the random vector <math>(X,Y)</math> satisfies
:<math>\varphi_{(X,Y)}(t,s) = \varphi_{X}(t)\cdot \varphi_{Y}(s).</math>

In particular the characteristic function of their sum is the product of their marginal characteristic functions:
:<math>\varphi_{X+Y}(t) = \varphi_X(t)\cdot\varphi_Y(t),</math>

though the reverse implication is not true. Random variables that satisfy the latter condition are called [[subindependence|subindependent]].