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Lecture 12 - Correlation

Lecture 12 - Correlation - 1 CORRELATION COEFFICIENT...

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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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