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slides_4_ranvecs

# Y e xy e x y e x y x y e xy x y x y x y e xy x y

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Y  E XY E X Y E X Y X Y E XY X Y X Y X Y E XY X Y where the last equality is useful for calculations. Note that Cov X , X Var X . By applying the Cauchy-Schwarz inequality to the deviations from means we get | Cov X , Y | SD X   SD Y 56

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(cov1) For random variables X and Y , Cov X , Y Cov Y , X . (cov2) For constants a 1 , b 1 , a 2 , and b 2 , Cov a 1 X b 1 , a 2 Y b 2 a 1 a 2 Cov X , Y (cov2) is liability for a measure of “association” because it implies that the strength of the association can be increased simply by multiplying one or both of the random variables by a large number. In particular, the covariance depends on units of measurement (such as dollars versus thousands of dollars or years versus months). 57
As a measure of association, it makes more sense to define the covariance between the standardized RVs, X X X and Y Y Y The covariance between the standardized variables is simply E  X X  Y Y  X Y which leads us to define the correlation between X and Y : Corr X , Y XY XY X Y Cov X , Y SD X SD Y 58

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(corr1) By the Cauchy-Schwarz inequality, 1 Corr X , Y 1 (corr2) For constants a 1 , b 1 , a 2 , and b 2 with a 1 a 2 0, Corr a 1 X b 1 , a 2 Y b 2 Corr X , Y These properties allow the magnitude of the correlation coefficient to have meaning. 59
If Corr X , Y 0 or, equivalently, Cov X , Y 0, we say X and Y are uncorrelated . Determining whether two variables are correlated – and the direction of the correlation – is the starting point of many studies. It remains to determine whether there is a a causal relationship, including whether other “confounding” factors affect both variables. 60

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EXAMPLE : It is easily established that amount of schooling ( X ) and annual income ( Y ) are positively correlated. Does going to school longer “cause” people to have higher earning power, or are education and income partly determined by a third factor such as “cognitive ability.” 61