If u and y have a bivariate normal distribution with zero me

If u and y have a bivariate normal distribution with zero means, show that cov(u^2, y^2) = 2[cov(u, u)]^2.

Solution

cov (u²,v²) = E(u²v²⁾ -E(u²)E(v²⁾

^{Cov (u,v) = E(UV)-E(u)E(V) = E(uv) as mean of u and v are 0}

Hence 2{ cov (u,v) }²  = 2{E(uv)}^2

But for any variable z, Var (z) = E(z^2) -{E(z)}^2

Hence cov  (u²,v²) = E(u²v²⁾ -E(u²)E(v²⁾ = E(u²v²⁾ -(varu)varv

=2{ cov (u,v) }²