Using the reduced proper motions of Tycho-2 stars with B-V from 0.75 to
  1.25: Monte Carlo simulations
  The stellar composition of the Tycho-2 Catalogue in the range $0.75<B-V<1.25$
has been reproduced through Monte Carlo simulations. For young and old stars of
the red giant clump (RGC), the red giant branch, subgiants, red dwarfs, and
thick-disk giants, we have specified the distributions in coordinates,
velocities, $B-V$, and $M_V$ as a function of $B-V$ and calculated their
reduced proper motions, photometric distances from the $(B-V)-M_V$ calibration,
and photoastrometric distances from the reduced proper motion -- $M_V$
calibration. Our simulations have shown the following: (1) a sample of
thin-disk giants within 500 pc with an admixture of less than 10\% of other
stars can be produced; (2) a sample of dwarfs within 100 pc almost without any
admixture of other stars can be produced; (3) the Local Spiral Arm affects the
RGC composition of any magnitude-limited catalog in favor of giants younger
than 2 Gyr; (4) the samples produced using reduced proper motions can be used
for kinematic studies, provided that the biases of the quantities being
determined are simulated and taken into account; (5) the photometric distances
correlate with the photoastrometric ones because of the correlation between the
proper motion and magnitude; (6) the photometric distances are closer to the
true ones for the red giant branch and red dwarfs as the categories of stars
with a clear $(B-V)-M_V$ relation, while the photoastrometric distances are
closer to the true ones for the RGC, subgiants, and thick-disk giants; (7) the
calculated distances differ systematically from the true ones, but they can be
used to analyze the three-dimensional distribution of stars. Our simulations
confirm the validity of our previous selection of RGC stars from Tycho-2.
