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Unformatted text preview: he units of the numerator, Cov(X, Y ). The situation
is similar to the use of the standardized versions of random variables X and Y , namely X −E [X ]
and Y −E [Y ] . These standardized versions have mean zero, variance one, and are dimensionless. In
fact, the covariance between the standardized versions of X and Y is ρX,Y :
Cov X − E [X ] Y − E [Y ]
σY = Cov XY
σX σY = Cov(X, Y )
= ρX,Y .
σX σY If the units of X or Y are changed (by linear or aﬃne scaling, such as changing from kilometers to
meters, or degrees C to degrees F) the correlation coeﬃcient does not change:
ρaX +b,cY +d = ρX,Y for a, c > 0. In a sense, therefore, the correlation coeﬃcient ρX,Y is the standardized version of the covariance,
Cov(X, Y ), or of the correlation, E [XY ]. As shown in the corollary of the following proposition,
correlation coeﬃcients are always in the interval [−1, 1]. As shown in Section 4.10, covariance or
correlation coeﬃcients play a central role for estimating Y by a linear...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
- Spring '08
- The Land