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Unformatted text preview: Noes on the correlation coefficient November 15, 2010 1 Introduction There are two types of correlation coefficients: the sample correlation coef- ficient, and the random variable analogue. Here, we will analyze and prove the properties of the random variable version; the properties for the sample version will be nearly identical, and follow from similar arguments. Given a sample ( X 1 , Y 1 ) , ..., ( X k , Y k ), the sample correlation coefficient is defined to be r := S XY √ S XX S Y Y , where for a sample ( U 1 , V 1 ) , ..., ( U k , V k ) we use the notation S UV = k summationdisplay i =1 ( U i − U )( V i − V ) . The random variable analogue is given by ρ := Cov( X, Y ) σ X σ Y , where σ 2 Z denotes the variance V ( Z ) of a random variable Z , and where Cov( X, Y ) denotes the covariance, defined to be Cov( X, Y ) := E (( X − μ X )( Y − μ Y )) = E ( XY ) − μ X μ Y . Note: In both cases, if the denominator in the definition of the correlation coefficient is 0, we will just say that the correlation coefficient is undefined . We have that ρ satisfies the following properties 1 1. − 1 ≤ ρ ≤ 1. r also satisfies this proerty....
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