  We consider the cubic nonlinear fourth-order Schr\"odinger equation \[
i\partial_t u - \Delta^2 u + \mu \Delta u = \pm |u|^2 u, \quad \mu \geq 0 \] on
$\mathbb{R}^N, N \geq 5$ with random initial data. We prove almost sure local
well-posedness below the scaling critical regularity. We also prove
probabilistic small data global well-posedness and scattering. Finally, we
prove the global well-posedness and scattering with a large probability for
initial data randomized on dilated cubes.
