# covariance

The covariance of two random variables $X_{1}$ and $X_{2}$ with mean (http://planetmath.org/ExpectedValue) $\mu_{1}$ and $\mu_{2}$ respectively is defined as

 $\mathrm{cov}(X_{1},X_{2}):=E[(X_{1}-\mu_{1})(X_{2}-\mu_{2})].$ (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:

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

Title covariance Covariance 2013-03-22 12:19:29 2013-03-22 12:19:29 Koro (127) Koro (127) 9 Koro (127) Definition msc 62-00 cov correlation correlation coefficient Variance