(a) Let Y1 and Y2 have a bivariate normal distribution. Show that the conditional distribution of Y1 given that Y2=y2 is a normal distribution with mean μ1+rhoσ2σ1(y2−μ2) and variance σ12(1−rho2). (10) (b) Consider the following bivariate joint probability density function for the random vector [XY] fX,Y(x,y)=2π1−rho2σxσy1exp[−2(1−rho2)1{σx2(x−μx)2−2rhoσxσy(x1−μx)(x2−μy) +σy2(x−μy)2}],−[infinity]0,σy>0, and ∣rho∣<1. Then fX,Y(x,y) is such that μ=(μx,μy)′Σ2×2=[σx2rhoσxσyrhoσxσyσy2]. Given that fY1,Y2(y1,y2)=k1exp[−21{2y12+y22+2y1y2−22y1−14y2+65}],−[infinity]
