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covariance (Definition)

The covariance of two random variables $ X_1$ and $ X_2$ with mean $ \mu_1$ and $ \mu_2$ respectively is defined as

$\displaystyle \mathrm{cov}(X_1,X_2) :=E[(X_1 - \mu_1)(X_2 - \mu_2)].$ (1)

The covariance of a random variable $ X$ with itself is simply the variance, $ E[(X - \mu)^2]$.

Covariance captures a measure of the correlation of two variables. Positive covariance indicates that as $ X_1$ increases, so does $ X_2$. Negative covariance indicates $ X_1$ decreases as $ X_2$ increases and vice versa. Zero covariance can indicate that $ X_1$ and $ X_2$ are uncorrelated.

The correlation coefficient provides a normalized view of correlation based on covariance:

$\displaystyle \mathrm{corr}(X,Y) :=\frac{\mathrm{cov}(X,Y)}{\sqrt{\mathrm{var}(X)\mathrm{var}(Y)}}.$ (2)

$ \mathrm{corr}(X,Y)$ ranges from -1 (for negatively correlated variables) through zero (for uncorrelated variables) to +1 (for positively correlated variables).

While if $ X$ and $ Y$ are independent we have $ \mathrm{corr}(X,Y)=0$, the latter does not imply the former.



"covariance" is owned by Koro. [ full author list (2) | owner history (1) ]
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See Also: variance

Other names:  cov, correlation, correlation coefficient
Keywords:  covariance, correlation coefficient
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Cross-references: imply, independent, negative, positive, variables, variance, random variables
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This is version 6 of covariance, born on 2002-02-13, modified 2004-03-21.
Object id is 1943, canonical name is Covariance.
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Classification:
AMS MSC62-00 (Statistics :: General reference works )

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