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The covariance of two random variables  and  with mean  and  respectively is defined as
![$\displaystyle \mathrm{cov}(X_1,X_2) :=E[(X_1 - \mu_1)(X_2 - \mu_2)].$](http://images.planetmath.org:8080/cache/objects/3360/l2h/img5.png "$\displaystyle \mathrm{cov}(X_1,X_2) :=E[(X_1 - \mu_1)(X_2 - \mu_2)].$") |
(1) |
The covariance of a random variable  with itself is simply the variance,
![$ E[(X - \mu)^2]$](http://images.planetmath.org:8080/cache/objects/3360/l2h/img7.png "$ E[(X - \mu)^2]$") .
Covariance captures a measure of the correlation of two variables. Positive covariance indicates that as  increases, so does  . Negative covariance indicates  decreases as  increases and vice versa. Zero covariance can indicate that  and  are uncorrelated.
The correlation coefficient provides a normalized view of correlation based on covariance:
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(2) |
 ranges from -1 (for negatively correlated variables) through zero (for uncorrelated variables) to +1 (for positively correlated variables).
While if  and  are independent we have
 , the latter does not imply the former.
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