Notice that this is a number without units it can be

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Notice that this is a number without units. It can be shown that 1 Corr( x, y ) 1. If Corr( x, y ) = 1, then there is a perfect positive correlation between x and y , which means that they lie on a straight line of positive slope. If Corr( x, y ) = 1, then there is a perfect negative correlation; and the straight line has a negative slope. In other cases, there is a 4
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scatter of points in the plane; and, if Corr( x, y ), then there is no linear relationship between x and y . These results concerning the range of the correlatison cofficient follow from a version of the Cauchy–Schwarz inequality, which will be established at the end of the next section. REGRESSION AND CONDITIONAL EXPECTATIONS Linear conditional expectations. If x, y are correlated, then a knowledge of one of them enables us to make a better prediction of the other. This knowledge can be used in forming conditional expectations. In some cases, it is reasonable to make the assumption that the conditional expectation E ( y | x ) is a linear function of x : E ( y | x ) = α + xβ. (i) This function is described as a linear regression equation. The error from predicting y by its conditional expectation can be denoted by ε = y E ( y | x ); and therefore we have y = E ( y | x ) + ε = α + + ε. Our object is to express the parameters α and β as functions of the moments of the joint probability distribution of x and y . Usually, the moments of the distribu- tion can be estimated in a straightforward way from a set of observations on x and y . Using the relationship that exits between the parameters and the theoretical moments, we should be able to find estimates for α and β corresponding to the estimated moments. We begin by multiplying equation (i) throughout by f ( x ), and by integrating with respect to x . This gives the equation E ( y ) = α + βE ( x ) , (ii) whence α = E ( y ) βE ( x ) . (iii) These equations shows that the regression line passes through the point E ( x, y ) = { E ( x ) , E ( y ) } which is the expected value of the joint distribution. By putting (iii) into (i), we find that E ( y | x ) = E ( y ) + β x E ( x ) , which shows how the conditional expectation of y differs from the unconditional expectation in proportion to the error of predicting x by taking its expected value. Now let us multiply (i) by x and f ( x ) and then integrate with respect to x to provide E ( xy ) = αE ( x ) + βE ( x 2 ) . (iv) Multiplying (ii) by E ( x ) gives E ( x ) E ( y ) = αE ( x ) + β E ( x ) 2 , (v) 5
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whence, on taking (v) from (iv), we get E ( xy ) E ( x ) E ( y ) = β E ( x 2 ) E ( x ) 2 , which implies that β = E ( xy ) E ( x ) E ( y ) E ( x 2 ) E ( x ) 2 = E x E ( x ) y E ( y ) E x E ( x ) 2 = C ( x, y ) V ( x ) . (vi) Thus, we have expressed α and β in terms of the moments E ( x ), E ( y ), V ( x ) and C ( x, y ) of the joint distribution of x and y . It should be recognised that the prediction error ε = y E ( y | x ) = y α is uncorrelated with the variable x . This is shown by writing E y E ( y | x ) x = E ( yx ) αE ( x ) βE ( x 2 ) = 0 , (vii) where the final equality comes from (iv). This result is readily intelligible; for, if the prediction error were correlated with the value of x , then we should not be using the information of x efficiently in predicting y .
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  • Spring '12
  • D.S.G.Pollock
  • Probability theory, probability density function, yj

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