Probability 2 - Notes 11 The bivariate and multivariate ...

Probability 2 - Notes 11 The bivariate and multivariate normal s(X,Y)have a bivariate normal distributionN( 1, 2, 21, 22, )if theirjoint isfX,Y(x,y) =12 1 2 (1 2)e 12(1 2)[(x 1 1)2 2 (x 1 1)(y 2 2)+(y 2 2)2](1)for allx,y. The parameters 1, 2may be any real numbers, 1>0, 2>0, and 1 is convenient to rewrite (1) in the formfX,Y(x,y) =ce 12Q(x,y),wherec=12 1 2 (1 2)andQ(x,y) = (1 2) 1[(x 1 1)2 2 (x 1 1)(y 2 2)+(y 2 2)2](2) marginal distributions ofN( 1, 2, 21, 22, )are normal with sXandYhaving density functionsfX(x) =1 2 1e (x 1)22 21,fY(y) =1 2 2e (y 2)22 expression (2) forQ(x,y)can be rearranged as follows:Q(x,y) =11 2[(x 1 1 y 2 2)2+(1 2)(y 2 2)2]=(x a)2(1 2) 21+(y 2)2 22,(3)wherea=a(y) = 1+ 1 2(y 2). HencefY(y) = fX,Y(x,y)dx=ce (y 2)22 22 e (x a)22(1 2) 21dx=1 2 2e (y 2)22 22,where the last step makes use of the formula e (x a)22 2dx= 2 with = 1 1 the formula forfX(x).

This is just the m.g.f. for the multivariate normal distribution with vector of means Am+b and variance-covariance matrix AVAT. Hence, from the uniqueness of the joint m.g.f, Y » N(Am+b;AVAT). Note that from (2) a subset of the Y0s is multivariate normal. NOTE. The results concerning the vector of means and variance-covariance matrix for linear

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  Normal, Bivariate

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