## Correlation (Statistical)

For two variables and ,

 (1)

where denotes Standard Deviation and is the Covariance of these two variables. For the general case of variables and , where , 2, ..., ,
 (2)

where are elements of the Covariance Matrix. In general, a correlation gives the strength of the relationship between variables. The variance of any quantity is alway Nonnegative by definition, so
 (3)

From a property of Variances, the sum can be expanded
 (4)

 (5)

 (6)

Therefore,
 (7)

Similarly,
 (8)

 (9)

 (10)

 (11)

Therefore,
 (12)

so . For a linear combination of two variables,
 (13)

Examine the cases where ,
 (14)

 (15)

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 .