For many-electron atoms, we discuss a generalized electron-pair density function g(q; a, b) that represents the probability density function for the magnitude \ar(i)+br(j)\ of the two-electron vector ar(i)+br(j) to be q, where a and b are real-valued parameters. It is pointed out that the function g(q; a, b) connects smoothly the single-electron density rho (r), the electron-pair intracule (relative motion) density h(r(12)), and the electron-pair extracule (center-of-mass motion) density d(R). Moreover, rho (r) is found to be a local extremum function of g(q; 1, b) with respect to the parameter b. Analogously, the single-electron moments [r(n)] are local extrema of the moments [q(n)]((a,b)) associated with g(q; a, b). An illustrative example is given for the helium atom within the Kellner approximation.