  In the low-energy effective theory of neutrinos, the Haar measure for unitary
matrices is very likely to give rise to the observed PMNS matrix. Assuming the
Haar measure, we determine the probability density functions for all quadratic,
quartic Majorana, and quartic Dirac rephasing invariants for an arbitrary
number of neutrino generations. We show that for a fixed number of neutrinos,
all rephasing invariants of the same type have the same probability density
function under the Haar measure. We then compute the moments of the rephasing
invariants to determine, with the help of the Mellin transform, the three
probability density functions. We finally investigate the physical implications
of our results in function of the number of neutrinos.
