Derivation of the Weizs\"acker Density Functional from Probability Theory

We demonstrate that the Weizs\"acker potential is an exact term in the universal functional in density functional theory (DFT) for the ground state of a system with $N$ electrons. This proof uses no approximations or physical arguments, and follows from the form of kinetic energy of the ground state and probability theory. We also examine the form of the other terms in the kinetic energy.