The Vision Thing Power Thirteen Bivariate Normal Distribution 1 Outline Circles around the origin Circles translated from the origin Horizontal ellipses around the translated origin Vertical ellipses around the translated origin Sloping ellipses 2 y x x 0 x2 1 y 0 y2 1 x y 0 3 y b a x a x2 1 x y b y2 1 x y 0 4 y x x 0 x2 y2 y 0 x y 0 5 y x x 0 x2 y2 y 0 x y 0 6 y b a x a x2 y2 x y b x y 0 7 y b a x a x2 y2 x y b x y 0 8 Why The Bivariate Normal Density and Circles f x y 1 2 x y exp 1 2 1 x x x 2 2 x x x y y y y y y 2 If means are zero and the variances are one and no correlation then f x y 1 2 exp 1 2 x2 y2 where f x y constant k for an isodensity ln2 k 1 2 x2 y2 and x2 y2 2ln2 k r2 9 Ellipses If x2 y2 f x y 1 2 x y exp 1 2 x x x 2 y y y 2 and x x x etc f x y 1 2 x y exp 1 2 x x 2 y y 2 where f x y constant k and ln k 2 x y 1 2 x x 2 y y 2 and x2 c2 y2 d2 1 is an ellipse 10 Correlation and Rotation of the Axes Y y X x x 0 x2 y2 y 0 x y 0 11 Bivariate Normal marginal conditional If x and y are independent then f x y f x f y i e the product of the marginal distributions f x and f y The conditional density function the density of y conditional on x f y x is the joint density function divided by the marginal density function of x f y x f x y f x 12 Conditional Distribution 2 2 y f y x 1 y 2 1 2 1 exp 1 2 1 y y x x y x the mean of the conditional distribution is y x x y x i e this is the expected value of y for a given value of x x x E y x x y x x y x The variance of the conditional distribution is VAR y x x x2 1 2 y Regression line intercept y x y x slope y x y x x a x2 y2 x y b x y 0 14 Bivariate Regression Another Perspective Regression line is the E y x line if y and x are bivariate normal intercept y x x y slope x y 15 Example Lab Six Rate of Return to GE stock 6 Series GE Sample 1993 01 1996 12 Observations 48 5 Mean Median Maximum Minimum Std Dev Skewness Kurtosis 4 3 2 1 Jarque Bera Probability 0 022218 0 019524 0 117833 0 058824 0 043669 0 064629 2 231861 1 213490 0 545122 0 0 05 0 00 0 05 0 10 16 Example Lab Six Rate of Return to S P500 Index 12 Series INDEX Sample 1993 01 1996 12 Observations 48 10 Mean Median Maximum Minimum Std Dev Skewness Kurtosis 8 6 4 2 Jarque Bera Probability 0 014361 0 017553 0 076412 0 044581 0 025430 0 453474 3 222043 1 743715 0 418174 0 0 04 0 02 0 00 0 02 0 04 0 06 0 08 17 Correlation Matrix GE INDEX GE 1 000000 0 636290 INDEX 0 636290 1 000000 18 Bivariate Regression Another Perspective Regression line is the E y x line if y and x are bivariate normal intercept y x x y slope x y y 0 022218 x 0 014361 x y 0 02543 0 043669 intercept 0 0064 slope 1 094 19 Returns Generating Process For GE Stock and S P 500 Index 0 15 0 10 GE 0 05 0 00 0 05 0 10 0 05 0 00 0 05 0 10 INDEX 20 Vs 0 0064 Vs 1 094 21 Bivariate Normal Distribution and the Linear probability Model 22 education Mean Educ Non Players Mean educ Players Non Players Players income mean income players Mean income non x a x2 y2 y b x y 0 23 education Mean Educ Non Players Mean educ Players Non Players Players income mean income players Mean income Non Players x a x2 y2 y b x y 0 24 education Mean Educ Non Players Mean educ Players Non Players Discriminating line Players income mean income players Mean income Non Players x a x2 y2 y b x y 0 25 Discriminant Function Linear Probability Function and Decision Theory Lab 6 Expected Costs of Misclassification E C C P N P P N P N C N P P N P P P Assume C P N C N P Relative Frequencies P N 23 100 1 4 P P 77 100 3 4 Equalize two costs of misclassification by setting fitted value of P P N i e Bern to 3 4 E C C P N 3 4 1 4 C N P 1 4 3 4 26 education Mean Educ Non Players Mean educ players Non Players Discriminating line Players income mean income players Mean income Non Players Note P P N is area of the non players distribution below southwest of the line x a x2 y2 y b x y 0 27 Set Bern 3 4 1 39 0 0216 education 0 0105 income solve for education as it depends on income and plot 28 7 non players misclassified as well as 14players misclassified 29 30 Decision Theory Moving the discriminant line I e changing the cutoff value from 0 75 to 0 5 changes the numbers of those misclassified favoring one population at the expense of another you need an implicit or explicit notion of the costs of misclassification such as C P N and C N P to make the necessary judgement of where to draw the line 31
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