Editing Uncorrelatedness (probability theory) (section)

===Example 2===
If <math>X</math> is a continuous random variable [[uniform distribution (continuous)|uniformly distributed]] on <math>[-1,1]</math> and <math>Y = X^2</math>, then <math>X</math> and <math>Y</math> are uncorrelated even though <math>X</math> determines <math>Y</math> and a particular value of <math>Y</math> can be produced by only one or two values of <math>X</math> :

<math> f_X(t)= {1 \over 2}  I_{[-1,1]} ;
 f_Y(t)= {1 \over {2 \sqrt{t}}} I_{]0,1]}</math>

on the other hand, <math> f_{X,Y}</math> is 0 on the triangle defined by <math>0<X<Y<1</math> although <math>f_X \times f_Y </math> is not null on this domain. 
Therefore <math> f_{X,Y} (X,Y) \neq f_X (X) \times f_Y (Y) </math> and the variables are not independent.

<math>
E[X] = {{1-1} \over 4} = 0 ; E[Y]= {{1^3 - (-1)^3}\over {3 \times 2} } = {1 \over 3} 
</math>

<math>
Cov[X,Y]=E \left [(X-E[X])(Y-E[Y]) \right ] = E  \left [X^3- {X \over 3} \right ] = {{1^4-(-1)^4}\over{4 \times 2}}=0
</math>

Therefore the variables are uncorrelated.

==When uncorrelatedness implies independence==
There are cases in which uncorrelatedness does imply independence.<!-- but not the only two --> One of these cases is the one in which both random variables are two-valued (so each can be linearly transformed to have a [[Bernoulli distribution]]).<ref>[http://www.math.uah.edu/stat/expect/Covariance.html Virtual Laboratories in Probability and Statistics: Covariance and Correlation], item 17.</ref>  Further, two jointly normally distributed random variables are independent if they are uncorrelated,<ref>{{cite book|chapter=Chapter 5.5 Conditional Expectation|pages=185–186|title=Introduction to Probability and Mathematical Statistics|year=1992|last1=Bain|first1=Lee|last2=Engelhardt|first2=Max|edition=2nd|isbn=0534929303}}</ref> although this does not hold for variables whose marginal distributions are normal and uncorrelated but whose joint distribution is not joint normal (see [[Normally distributed and uncorrelated does not imply independent]]).

==Generalizations==
===Uncorrelated random vectors===
Two [[random vector]]s <math>\mathbf{X}=(X_1,\ldots,X_m)^T </math> and <math>\mathbf{Y}=(Y_1,\ldots,Y_n)^T </math> are called uncorrelated if
:<math>\operatorname{E}[\mathbf{X} \mathbf{Y}^T] = \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{Y}]^T</math>.

They are uncorrelated if and only if their [[cross-covariance matrix]] <math>\operatorname{K}_{\mathbf{X}\mathbf{Y}}</math> is zero.<ref name=Gubner>{{cite book |first=John A. |last=Gubner |year=2006 |title=Probability and Random Processes for Electrical and Computer Engineers |publisher=Cambridge University Press |isbn=978-0-521-86470-1}}</ref>{{rp|p.337}}

Two complex random vectors <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> are called '''uncorrelated''' if their cross-covariance matrix and their pseudo-cross-covariance matrix is zero, i.e. if
:<math>\operatorname{K}_{\mathbf{Z}\mathbf{W}}=\operatorname{J}_{\mathbf{Z}\mathbf{W}}=0</math>
where
:<math> \operatorname{K}_{\mathbf{Z}\mathbf{W}} =\operatorname{E}[(\mathbf{Z}-\operatorname{E}[\mathbf{Z}]){(\mathbf{W}-\operatorname{E}[\mathbf{W}])}^{\mathrm H}]</math>
and
:<math> \operatorname{J}_{\mathbf{Z}\mathbf{W}} =\operatorname{E}[(\mathbf{Z}-\operatorname{E}[\mathbf{Z}]){(\mathbf{W}-\operatorname{E}[\mathbf{W}])}^{\mathrm T}]</math>.