Given that $X$ is random matrix ($X'$ is transposed matrix) and $\Omega$ is invertible square matrix is inequality $$EX'X\cdot(EX'\Omega X)^{-1}\cdot EX'X\leq EX'\Omega^{-1}X$$ correct? All the expectations as well as inverse of $EX'\Omega X$ exist.
Maybe (but it is probably wrong)
$$EX'X\cdot(EX'\Omega X)^{-1}\cdot EX'X \leq EX'X\cdot(X'\Omega X)^{-1}\cdot X'X $$
Then I somehow need to show that
$$X'X\cdot(X'\Omega X)^{-1}\cdot X'X = X'\Omega^{-1}X$$
Maybe expressing $\Omega$ as $\Omega^{1/2}\cdot \Omega^{-1/2}$ might somehow work.
