Transcription of General Bivariate Normal - Duke University
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General Bivariate Normal - Duke University
Lecture 22: Bivariate Normal DistributionStatistics 104 Colin RundelApril 11, DistributionsGeneral Bivariate NormalLetZ1,Z2 N(0,1), which we will use to build a General Bivariate (z1,z2) =12 exp[ 12(z21+z22)]We want to transform these unit Normal distributions to have the followarbitrary parameters: X, Y, X, Y, X= XZ1+ XY= Y[ Z1+ 1 2Z2] + YStatistics 104 (Colin Rundel)Lecture 22 April 11, 20121 / DistributionsGeneral Bivariate Normal - MarginalsFirst, lets examine the marginal distributions ofXandY,X= XZ1+ X= XN(0,1) + X=N( X, 2X)Y= Y[ Z1+ 1 2Z2] + Y= Y[ N(0,1) + 1 2N(0,1)] + Y= Y[N(0, 2) +N(0,1 2)] + Y= YN(0,1) + Y=N( Y, 2Y)Statistics 104 (Colin Rundel)Lecture 22 April 11, 20122 / DistributionsGeneral Bivariate Normal - Cov/CorrSecond, we can findCov(X,Y) and (X,Y)Cov(X,Y) =E[(X E(X))(Y E(Y))]=E[( XZ1+ X X)( Y[ Z1+ 1 2Z2] + Y Y)]=E[( XZ1)( Y[ Z1+ 1 2Z2])]= X YE[ Z21+ 1 2Z1Z2]= X Y E[Z21]= X Y (X,Y) =Cov(X,Y)
7 7 5 Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 5 / 22 6.5 Conditional Distributions General Bivariate Normal - Density The rst thing we need to nd are the inverses of the …
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