The mean value and the variance give important information about the distribution for a real random variable X. We consider the expectation of an appropriate function of a pair (X, Y) which gives useful information about their joint distribution. This is the covariance function.
Covariance and the correlation coefficient
The mean value
and the variance
give
important information about the distribution for real random variable
X . Can
the expectation of an appropriate function of
give useful information about
the joint distribution? A clue to one possibility is given in the expression
The expression
vanishes if the pair is independent (and in some
other cases). We note also that for
and
To see this, expand the expression
and use linearity to get
which reduces directly to the desired expression.
Now for given
ω ,
is the variation of
X from its mean
and
is the variation of
Y from its mean. For this reason,
the following terminology is used.
Definition . The quantity
is called
the
covariance of
X and
Y .
If we let
and
be the centered random
variables, then
Note that the variance of
X is the covariance of
X with itself.
If we standardize, with
and
,
we have
Definition . The
correlation coefficient
is the quantity
Thus
. We examine these concepts for information on the joint distribution.
By Schwarz' inequality (E15), we have
Now equality holds iff
We conclude
, with
iff
Relationship between
ρ and the joint distribution
We consider first the distribution for the standardized pair
Since
we obtain the results for the distribution for
by the mapping
Joint distribution for the standardized variables
,
iff
iff all probability mass is on the line
.
iff
iff all probability mass is on the line
.
If
, then at least some of the mass must fail to be on these lines.
The
lines for the
distribution are:
Consider
. Then
. Reference to
[link] shows this is the average of the
square of the distancesof the points
from the line
(i.e., the variance about
the line
). Similarly for
,
is the variance about
. Now
Thus
is the variance about
(the
line)
is the variance about
(the
line)
Now since
the condition
is the condition for equality of the two variances.
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