The covarianceMathworldPlanetmath of two random variablesMathworldPlanetmath X1 and X2 with mean ( μ1 and μ2 respectively is defined as

cov(X1,X2):=E[(X1-μ1)(X2-μ2)]. (1)

The covariance of a random variable X with itself is simply the varianceMathworldPlanetmath, E[(X-μ)2].

Covariance captures a measure of the correlationMathworldPlanetmath of two variables. Positive covariance indicates that as X1 increases, so does X2. Negative covariance indicates X1 decreases as X2 increases and vice versa. Zero covariance can indicate that X1 and X2 are uncorrelated.

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

corr(X,Y):=cov(X,Y)var(X)var(Y). (2)

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 independentPlanetmathPlanetmath we have corr(X,Y)=0, the latter does not imply the former.

Title covariance
Canonical name Covariance
Date of creation 2013-03-22 12:19:29
Last modified on 2013-03-22 12:19:29
Owner Koro (127)
Last modified by Koro (127)
Numerical id 9
Author Koro (127)
Entry type Definition
Classification msc 62-00
Synonym cov
Synonym correlation
Synonym correlation coefficient
Related topic Variance