Cube Line Picking--Face and Interior

Consider the distribution of distances  between a point picked at random in the interior of a unit cube and on a face of the cube. The probability function, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form

(1)

The first even raw moments  for , 2, 4, ... are 1, 2/3, 11/18, 211/315, 187/225, 11798/10395, ....

REFERENCES:

Mathai, A. M.; Moschopoulos, P.; and Pederzoli, G. "Distance between Random Points in a Cube." J. Statistica 59, 61-81, 1999.