  We present a method to characterize statistically the parameters of a
detached binary sample - binary fraction, separation distribution, and mass
ratio distribution - using noisy radial-velocity data with as few as two,
randomly spaced, epochs per object. To do this, we analyze the distribution of
DRVmax, the maximum radial-velocity difference between any two epochs for the
same object. At low values, the core of this distribution is dominated by
measurement errors, but for large enough samples there is a high-velocity tail
that can effectively constrain the parameters of the binary population. We
discuss our approach for the case of a population of detached white-dwarf (WD)
binaries with separations that are decaying via gravitational wave emission. We
derive analytic expressions for the present-day distribution of separations,
integrated over the star-formation history of the Galaxy, for parametrized
initial WD separation distributions at the end of the common-envelope phase. We
use Monte Carlo techniques to produce grids of simulated DRVmax distributions
with specific binary population parameters, and the same sampling cadences and
radial velocity errors as the observations, and we compare them to the real
DRVmax distribution to constrain the properties of the binary population. We
illustrate the sensitivity of the method to both the model and the
observational parameters. In the particular case of binary white dwarfs, every
model population predicts a merger rate per star which can easily be compared
to type-Ia supernova rates. In a companion paper, we apply the method to a
sample of about 4000 WDs from the Sloan Digital Sky Survey, and we find a
merger rate remarkably similar to the rate of Type-Ia supernovae in
Milky-Way-like galaxies.
