1. 1 The Vision Thing Power Thirteen Bivariate Normal Distribution

2. 2 Outline Circles around the origin Circles translated from the origin Horizontal ellipses around the (translated) origin Vertical ellipses around the (translated) origin Sloping ellipses

3. 3 x y  x = 0,  x 2 =1  y = 0,  y 2 =1  x, y = 0

4. 4 x y  x = a,  x 2 =1  y = b,  y 2 =1  x, y = 0 a b

5. 5 x y  x = 0,  x 2 >  y 2  y = 0  x, y = 0

6. 6 x y  x = 0,  x 2 <  y 2  y = 0  x, y = 0

7. 7 x y  x = a,  x 2 >  y 2  y = b  x, y > 0 a b

8. 8 x y  x = a,  x 2 >  y 2  y = b  x, y < 0 a b

9. 9 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 )*(x 2 + y 2 ), where f(x,y) = constant, k, for an isodensity ln2  k =(-1/2)*(x 2 + y 2 ), and (x 2 + y 2 )= -2ln2  k=r 2

10. 10 Ellipses If  x 2 >  y 2, 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 x 2 /c 2 + y 2 /d 2 = 1 is an ellipse

11. 11 x y  x = 0,  x 2 <  y 2  y = 0  x, y < 0 Correlation and Rotation of the Axes Y’ X’

12. 12 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)

13. Conditional Distribution f(y/x)= 1/[  y ]exp{[-1/2(1-    y 2 ]* [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*) =  x 2 (1-  ) 2nn

14. 14 x y  x = a,  x 2 >  y 2  y = b  x, y > 0 xx yy Regression line intercept:  y -  x (  y /  x ) slope:  (  y /  x )

15. 15 Bivariate Normal Distribution and the Linear probability Model

16. 16 income education  x = a,  x 2 >  y 2  y = b  x, y > 0 mean income non Mean educ. non Mean Educ Players Mean income Players Players Non-players

17. 17 income education  x = a,  x 2 >  y 2  y = b  x, y > 0 mean income non Mean educ. non Mean Educ Players Mean income Players Players Non-players

18. 18 income education  x = a,  x 2 >  y 2  y = b  x, y > 0 mean income non Mean educ. non Mean Educ Players Mean income Players Players Non-players Discriminating line