The Vision Thing Power ThirteenOutlineSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Why? The Bivariate Normal Density and CirclesEllipsesSlide 11Bivariate Normal: marginal & conditionalConditional DistributionSlide 14Bivariate Normal Distribution and the Linear probability ModelSlide 16Slide 17Slide 181The Vision ThingPower ThirteenBivariate Normal Distribution2Outline•Circles around the origin•Circles translated from the origin•Horizontal ellipses around the (translated) origin•Vertical ellipses around the (translated) origin•Sloping ellipses3xyx = 0, x2 =1y = 0, y2 =1x, y = 04xyx = a, x2 =1y = b, y2 =1x, y = 0ab5xyx = 0, x2 > y2 y = 0x, y = 06xyx = 0, x2 < y2 y = 0x, y = 07xyx = a, x2 > y2 y = bx, y > 0ab8xyx = a, x2 > y2 y = bx, y < 0ab9Why? The Bivariate Normal Density and Circles•f(x, y) = {1/[2xy]}*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•ln2k =(-1/2)*(x2 + y2), and (x2 + y2)= -2ln2k=r210Ellipses•If x2 > y2, f(x,y) = {1/[2xy]}*exp{(-1/2)* ([(x-x)/x]2 + ([(y-y)/y]2}, and x# = (x-x) etc.•f(x,y) = {1/[2xy]}exp{(-1/2)* ([x#/x]2 + [y#/y]2) , where f(x,y) =constant, k, and ln{k [2xy]} = (-1/2) ([x#/x]2 + [y#/y]2 )and x2/c2 + y2/d2 = 1 is an ellipse11xyx = 0, x2 < y2 y = 0x, y < 0Correlation and Rotation of the Axes Y’X’12Bivariate 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)Conditional Distribution•f(y/x)= 1/[y ]exp{[-1/2(1-y2]* [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-)2nn 2/12)1(214xyx = a, x2 > y2 y = bx, y > 0xyRegression lineintercept:y - x(y/x)slope:(y/x)15Bivariate Normal Distribution and the Linear probability Model16incomeeducationx = a, x2 > y2 y = bx, y > 0mean income nonMeaneduc.nonMeanEducPlayersMean income PlayersPlayersNon-players17incomeeducationx = a, x2 > y2 y = bx, y > 0mean income nonMeaneduc.nonMeanEducPlayersMean income PlayersPlayersNon-players18incomeeducationx = a, x2 > y2 y = bx, y > 0mean income nonMeaneduc.nonMeanEducPlayersMean income
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