Editing Independence (probability theory) (section)

===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}}