September 12th, 2008, 9:31 am
we assume x, e1, e2 ~ N(0,1) and also assume E(x.e1)=E(x.e2)=0y=x.p1+e1.p1' because correlation(x,y) = p1 and where p1' = sqrt(1-p1^2)z=x.p2+e2.p2' because correlation(x,z) = p2 and where p2' = sqrt(1-p2^2)=> y, z ~ N(0,1) you could easily verify this with simple assumptions aboveNow as correlation between y and z = E(y.z) = E(p1.p2.x^2+e1.e2.p1'.p2'+something+something) = p1.p2 + p1'.p2'.E(e1.e2)+0+0As we know -1<E(e1.e2)<1Aaron's result follows