Needle-scattering analyzer moments

For isotropically oriented independent needles illuminated with $z$ polarization and viewed along $y$, the amplitudes through $x$ and $z$ analysers are proportional to $u_x{u}_z$ and $u_z^2$. The moments $⟨u_x^2{u}_z^2⟩=1/15$ and $⟨u_z^4⟩=1/5$ give intensity ratio $1:3$.