The statistical correlation of two real random variables 
and 
with finite positive variances is defined by
 |
(1)
|
where 
denotes standard deviation and 
is the covariance of
these two variables. For the general case of variables 
and 
with finite positive variances,
where 
, 2, ..., 
,
 |
(2)
|
where 
are elements of the covariance matrix. In general,
a correlation gives the strength of the linear relationship between variables. For
,
 |
(3)
|
The variance of any quantity is always nonnegative by definition, so
 |
(4)
|
From a property of variances, the sum can be expanded
 |
(5)
|
 |
(6)
|
 |
(7)
|
Therefore,
 |
(8)
|
Similarly,
 |
(9)
|
 |
(10)
|
 |
(11)
|
 |
(12)
|
Therefore,
 |
(13)
|
so 
.
For a real number 
, the variance of a linear combination of the variables satisfies
 |  |  |
(14)
|
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
Examine the cases where 
,
 |
(18)
|
 |
(19)
|
The variance is zero for 
, so 
is constant almost surely.
Thus 
almost surely for some constant 
. The variables are perfectly correlated when 
and perfectly anticorrelated when 
.
