  The Kepler Mission is uniquely suited to study the frequencies of extrasolar
planets. This goal requires knowledge of the incidence of false positives such
as eclipsing binaries in the background of the targets, or physically bound to
them, which can mimic the photometric signal of a transiting planet. We perform
numerical simulations to predict the occurrence of astrophysical false
positives detectable by the Mission, and quantify the fraction of them that
would pass the Kepler candidate vetting procedure. By comparing their
distribution with that of the Kepler Objects of Interest detected during the
first six quarters of operation of the spacecraft, we infer the false positive
rate of Kepler and study its dependence on spectral type, candidate planet
size, and orbital period. We find that the global false positive rate of Kepler
is 9.4 %, peaking for giant planets (6-22 Earth radii) at 17.7 %, reaching a
low of 6.7 % for small Neptunes (2-4 Earth radii), and increasing again for
Earth-size planets (0.8-1.25 Earth radii) to 12.3 %.
  Most importantly, we also quantify and characterize the distribution and rate
of occurrence of planets down to Earth size with no prior assumptions on their
frequency, by subtracting from the population of actual Kepler candidates our
simulated population of astrophysical false positives. We find that 16.5 +/-
3.6 % of main-sequence FGK stars have at least one planet between 0.8 and 1.25
Earth radii with orbital periods up to 85 days. There is no significant
dependence of the rates of planet occurrence between 0.8 and 4 Earth radii with
spectral type. In the process, we derive also a prescription for the signal
recovery rate of Kepler that enables a good match to both the KOI size and
orbital period distribution, as well as their signal-to-noise distribution.
