dropping the explicit time dependence notation for clarity. To find the MSD, one can take one of two paths: one can explicitly calculate ⟨ ⟨ x 2 ⟩ ⟩ and ⟨ ⟨ x ⟩ ⟩ , then plug the result back into the definition of the MSD; or one could find the moment-generating function, an extremely useful, and general function when dealing with probability densities. The moment-generating function describes the k th moment of the PDF. The first moment of the displacement PDF shown above is simply the mean: ⟨ ⟨ x ⟩ ⟩ . The second moment is given as ⟨ ⟨ x 2 ⟩ ⟩ .