For many-electron atoms, the generalized electron-pair density function g(q;a,b) represents the probability density function for the magnitude \ ar(i)+br(j)\ of two-electron vector ar(i)+br(j) to be q, where a and b are real-valued parameters. It is pointed out that the second moments (q(2)) ((a,b)), associated with g(q;a,b), are related with several physical properties such as the diamagnetic susceptibility chi (d), the form factor F(mu), the incoherent scattering function S(mu), the minus first moment M-1 of the oscillator strength density, and the dipole polarizability alpha. The corresponding moments (t(2)) ((a,b)) in momentum space are connected with the mass polarization correction epsilon (M), the first moment M-1 of the oscillator strength density, and the nuclear momentum squared (p(0)(2)) as well as the electronic kinetic energy 1/2(Sigma (i)p(i)(2)). The average excitation energy DeltaE is also estimated from (q(2)) ((a,b)) and (t(2)) ((a,b)).