# barcode font reporting services g(x) having property (37) is called a least-squares regression curve of Y on X. We have in Software Printer QR Code ISO/IEC18004 in Software g(x) having property (37) is called a least-squares regression curve of Y on X. We have

g(x) having property (37) is called a least-squares regression curve of Y on X. We have
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Theorem 8-1 If X and Y are random variables having joint density function or probability function f(x, y), then there exists a least-squares regression curve of Y on X, having property (37), given by y g(x) E(Y Z X x) (38)
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provided that X and Y each have a variance that is finite. Note that E(Y Z X x) is the conditional expectation of Y given X x, as defined on page 82. Similar remarks can be made for a least-squares regression curve of X on Y. In that case, (37) is replaced by E5[X h(Y)]26 a minimum
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and (38) is replaced by x h( y) E(X Z Y y). The two regression curves y g(x) and x h( y) are different in general. An interesting case arises when the joint distribution is the bivariate normal distribution given by (49), page 117. We then have the following theorem: Theorem 8-2 If X and Y are random variables having the bivariate normal distribution, then the least-squares regression curve of Y on X is a regression line given by y mY sY r ra x mX sX b (39)
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where
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sXY sX sY
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(40)
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represents the population correlation coefficient. We can also write (39) as y where mY b b(x sXY s2 X mX) (41) (42)
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Similar remarks can be made for the least-squares regression curve of X on Y, which also turns out to be a line [given by (39) with X and Y, x and y, interchanged]. These results should be compared with corresponding ones on page 268.
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CHAPTER 8 Curve Fitting, Regression, and Correlation
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In case f (x, y) is not known, we can still use the criterion (37) to obtain approximating regression curves for the population. For example, if we assume g(x) a bx, we obtain the least-squares regression line (39), where a, b are given in terms of the (unknown) parameters mX, mY, sX, sY, r. Similarly if g(x) a bx gx2, we can obtain a least-squares regression parabola, etc. See Problem 8.39. In general, all of the remarks made on pages 266 to 271 for samples are easily extended to the population. For example, the standard error of estimate in the case of the population is given in terms of the variance and correlation coefficient by s2 Y.X which should be compared with (25), page 269. s2 (1 Y r2) (43)
Probability Interpretation of Correlation
From the above remarks it is clear that a population correlation coefficient should provide a measure of how well a given population regression curve fits the population data. All the remarks previously made for correlation in a sample apply as well to the population. For example, if g(x) is determined by (37), then E[(Y # Y )2] E[(Y Yest)2] E[(Yest # Y )2] (44)
# where Yest g(X ) and Y E(Y ). The three quantities in (44) are called the total, unexplained, and explained variations, respectively. This leads to the definition of the population correlation coefficient r, where r2 explained variation total variation E[(Yest E[(Y # Y )2] # Y )2] (45)
For the linear case this reduces to (40). Results similar to (31) through (34) can also be written for the case of a population and linear regression. The result (45) is also used to define r, in the nonlinear case.
Sampling Theory of Regression
The regression equation y a bx is obtained on the basis of sample data. We are often interested in the corresponding regression equation y a bx for the population from which the sample was drawn. The following are some tests concerning a normal population. To keep the notation simple, we shall follow the common convention of indicating values of sampling random variables rather than the random variables themselves. 1. TEST OF HYPOTHESIS B 5 b. To test the hypothesis that the regression coefficient b is equal to some specified value b, we use the fact that the statistic t b b !n sy.x >sx 2 (46)
has Student s distribution with n 2 degrees of freedom. This can also be used to find confidence intervals for population regression coefficients from sample values. See Problems 8.43 and 8.44. 2. TEST OF HYPOTHESES FOR PREDICTED VALUES. Let y0 denote the predicted value of y corresponding to x x0 as estimated from the sample regression equation, i.e., y0 a bx0. Let yp denote the predicted value of y corresponding to x x0 for the population. Then the statistic t ( y0 sy.x 2n 1 yp)!n [n(x0 2 x)2 >s2] # x (47)
has Student s distribution with n 2 degrees of freedom. From this, confidence limits for predicted population values can be found. See Problem 8.45. 3. TEST OF HYPOTHESES FOR PREDICTED MEAN VALUES. Let y0 denote the predicted value of y corresponding to x x0 as estimated from the sample regression equation, i.e., y0 a bx0. Let yp #