  Time delays between lensed multiple images have been known to provide an
interesting probe of the Hubble constant, but such application is often limited
by degeneracies with the shape of lens potentials. We propose a new statistical
approach to examine the dependence of time delays on the complexity of lens
potentials, such as higher-order perturbations, non-isothermality, and
substructures. Specifically, we introduce a reduced time delay and explore its
behavior as a function of the image configuration that is characterized by the
asymmetry and opening angle of the image pair. In particular we derive a
realistic conditional probability distribution. We find that the probability
distribution is sensitive to the image configuration such that more asymmetric
and/or smaller opening angle image pairs are more easily affected by
perturbations on the primary lens potential. On average time delays of double
lenses are less scattered than those of quadruple lenses. Furthermore, the
realistic conditional distribution allows a new statistical method to constrain
the Hubble constant. We find that 16 published time delay quasars constrain the
Hubble constant to be H_0=70+/-6 km/s/Mpc, where the value and its error are
estimated using jackknife resampling. After including rough estimates of the
sizes of important systematic errors, we find H_0=68+/-6(stat.)+/-8(syst.)
km/s/Mpc. The reasonable agreement with other estimates indicates the
usefulness of our new approach as a cosmological and astrophysical probe,
particularly in the era of large-scale synoptic surveys. (Abridged)
