1
CORRELATION COEFFICIENT
Lecture 12
ORIE3500/5500 Summer2011 Li
1
Correlation Coefficient
The covariance of
X
and
Y
gives us some idea about whether there is any
linear relationship between
X
and
Y
. But it is difficult to judge how strong
the relation is, since there is no proper scale in which we can measure it. So
we scale it to 1 and define correlation coeffiient. Correlation coefficient of
X
and
Y
is defined to be
corr
(
X, Y
) =
ρ
X,Y
=
cov
(
X, Y
)
p
var
(
X
)
p
var
(
Y
)
=
cov
(
X, Y
)
σ
X
σ
Y
.
The covariance is robust with change of location, that is,
cov
(
X
+
a, Y
+
b
) =
cov
(
X, Y
)
.
But it changes if we saw earlier that scaling the random variables changes
covariance,
cov
(
aX, bY
) =
ab
·
cov
(
X, Y
)
.
In order to measure linear relationship between random variables, this is
not a desirable property. We will not be able to infer a strong positive linear
relationship between
X
and
Y
, even if the covariance is a large number, since
a bigger scaling or higher dispersion may be a reason for that. So dividing
the covariance by the dispersion of
X
and
Y
gives us a better understanding
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 Spring '11
 SONG
 Vector Space, Correlation, Discrete probability distribution, Covariance and correlation

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