  Planets on eccentric orbits have a higher geometric probability of transiting
their host star. By application of Bayes' theorem, we reverse this logic to
show that the eccentricity distribution of transiting planets is positively
biased. Adopting the flexible Beta distribution as the underlying prior for
eccentricity, we derive the marginalized transit probability as well as the
a-priori joint probability distribution of eccentricity and argument of
periastron, given that a planet is known to transit. These results allow to
demonstrate that most planet occurrence rate calculations using Kepler data
have overestimated the prevalence of planets by ~10%. Indeed, the true
occurrence of planets from transit surveys is fundamentally intractable without
a prior assumption for the eccentricity distribution. Further more, we show
that previously extracted eccentricity distributions using Kepler data are
positively biased. In cases where one wishes to impose an informative
eccentricity prior, we provide a recursive algorithm to apply inverse transform
sampling of our joint prior probability distribution. Computer code of this
algorithm, ECCSAMPLES, is provided to enable the community to sample directly
from the prior.
