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Covariance and correlation are related parameters that indicate the extent to which two random variables co-vary. Suppose there are two technology stocks. If they are affected by the same industry trends, their prices will tend to rise or fall together. They co-vary. Covariance and correlation measure such a tendency. Let's formalize this. Consider a random vector X
whose components are random variables
|
Given any pair of components,
and
,
we denote their covariance as either
or
.
The covariance is defined by the
expectation
 |
[2] |
where
and
are the means of
and
.
By definition, covariance is symmetric, with
.
Also, the covariance of any component
with itself is that component’s
variance:
 |
[3] |
We summarize all the covariance's of a random vector X with a covariance matrix:
 |
[4] |
Due to the symmetry property of covariances, this is necessarily a symmetric matrix. It can be shown that covariance matrices are positive definite or positive semidefinite.
The magnitude of a covariance
depends upon the standard deviations of the two components
and
.
To obtain a more direct indication of how two components co-vary, we scale
covariance to obtain correlation.
Given any pair of components,
and
,
we denote their correlation as either
or
.
The correlation is defined as
 |
[5] |
where
and
are the standard deviations of
and
.
By construction, a correlation is always a
number between –1 and 1. Correlation inherits the symmetry property of
covariance:
.
From [3] and [5],
,
which indicates that a random variable co-varies perfectly with itself. If
and
are independent, their correlation is 0. The converse is not true. As with
covariances, we can summarize all the correlations of a random vector
X with a symmetric correlation
matrix:
 |
[6] |
Like covariance matrices, correlation matrices must be positive definite or positive semidefinite.
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