For statistically uncorrelated wave functions, the electron density p(r) is known to be an extremum function of the generalized electron-pair density g(q; a, b), which smoothly connects p(r), the electron-pair intracule (relative motion) density h(u), and the electron-pair extracule (center-of-mass motion) density d(R). The present systematic examination of the numerical Hartree-Fock results of moments (r(n)), (u(n)), (R-n) and density values at the origin p(0), h(0), d(0) for 102 neutral atoms and 96 singly charged ions suggests that p(u) is Focal maximum for a small q and is local minimum for a large q of the function g(q; a, b). Analogous results ar e obtained for the momentum-space counterparts Pi (p) and (g) over bar (t; a, b).