Editing Covariance (section)

===Uncorrelatedness and independence===
{{main|Correlation and dependence}}
Random variables whose covariance is zero are called [[uncorrelated]].<ref name=KunIlPark/>{{rp|p= 121}} Similarly, the components of random vectors whose [[covariance matrix]] is zero in every entry outside the main diagonal are also called uncorrelated.

If <math>X</math> and <math>Y</math> are [[statistical independence|independent random variables]], then their covariance is zero.<ref name=KunIlPark/>{{rp|p= 123}}<ref>{{Cite web | url=http://www.randomservices.org/random/expect/Covariance.html| title=Covariance and Correlation | last=Siegrist|first=Kyle| publisher=University of Alabama in Huntsville |access-date=Oct 3, 2022}}</ref> This follows because under independence,
<math display="block">\operatorname{E}[XY] = \operatorname{E}[X] \cdot \operatorname{E}[Y]. </math>

The converse, however, is not generally true. For example, let <math>X</math> be uniformly distributed in <math>[-1,1]</math> and let <math>Y = X^2</math>. Clearly, <math>X</math> and <math>Y</math> are not independent, but
<math display="block">\begin{align}
  \operatorname{cov}(X, Y) &= \operatorname{cov}\left(X, X^2\right) \\
         &= \operatorname{E}\left[X \cdot X^2\right] - \operatorname{E}[X] \cdot \operatorname{E}\left[X^2\right] \\
         &= \operatorname{E}\left[X^3\right] - \operatorname{E}[X]\operatorname{E}\left[X^2\right] \\
         &= 0 - 0 \cdot \operatorname{E}[X^2] \\
         &= 0.  
\end{align}</math>

In this case, the relationship between <math>Y</math> and <math>X</math> is non-linear, while correlation and covariance are measures of linear dependence between two random variables. This example shows that if two random variables are uncorrelated, that does not in general imply that they are independent. However, if two variables are [[Multivariate normal distribution|jointly normally distributed]] (but not if they are merely [[Normally distributed and uncorrelated does not imply independent|individually normally distributed]]), uncorrelatedness ''does'' imply independence.<ref>{{Cite book |title=A modern introduction to probability and statistics: understanding why and how |date=2005 |publisher=Springer |isbn=978-1-85233-896-1 |editor-last=Dekking |editor-first=Michel |series=Springer texts in statistics |location=London [Heidelberg]}}</ref>

<math>X</math> and <math>Y</math> whose covariance is positive are called positively correlated, which implies if <math>X>E[X]</math> then likely <math>Y>E[Y]</math>. Conversely, <math>X</math> and <math>Y</math> with negative covariance are negatively correlated, and if <math>X>E[X]</math> then likely <math>Y<E[Y]</math>.