  In this paper we will study the statistics of the unit geodesic flow normal
to the boundary of a hyperbolic manifold with non-empty totally geodesic
boundary. Viewing the time it takes this flow to hit the boundary as a random
variable, we derive a formula for its moments in terms of the orthospectrum.
The first moment gives the average time for the normal flow acting on the
boundary to again reach the boundary, which we connect to Bridgeman's identity
(in the surface case), and the zeroth moment recovers Basmajian's identity.
Furthermore, we are able to give explicit formulae for the first moment in the
surface case as well as for manifolds of odd dimension. In dimension two, the
summation terms are dilogarithms. In dimension three, we are able to find the
moment generating function for this length function.
