Statistical Correlation

For two random variates and , the correlation is defined bY

where denotes standard deviation and is the covariance of these two variables. For the general case of variables and , where , 2, ..., ,

where are elements of the covariance matrix. In general, a correlation gives the strength of the relationship between variables. For ,

The variance of any quantity is always nonnegative by definition, so

From a property of variances, the sum can be expanded

Therefore,

Similarly,

Therefore,

so .

For a linear combination of two variables,

Examine the cases where ,

The variance will be zero if , which requires that the argument of the variance is a constant. Therefore, , so . If , is either perfectly correlated () or perfectly anticorrelated () with .