2 - People.stat.sfu.ca

Since p 2 is a special case of p 1 with ρ = 0 we find
which is not
unless ρ = 0.
E(X 2 Y 2 ) = [1 + 2ρ 2 + 1]/2 = 1 + ρ 2
E ρ/2 (X 2 Y 2 ) = 1 + 2(ρ/2) 2 = 1 + ρ 2 /2
(b) Generalize the construction to show that there rv’s X and Y such
that X and Y are each standard normal, X and Y are uncorrelated
but X and Y are not independent.
Define p 1 as in the first part and p 2 like p 1 but with correlation
−ρ. The density p = (p 1 + p 2 )/2 has standard normal margins
and correlation 0 but does not factor. In this case
E(X 2 Y 2 ) = 1 + 2ρ 2 ≠ 1
unless ρ = 0. Here is a contour plot of the joint density when
ρ = 0.8.
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