  We apply standard Markov chain Monte Carlo (MCMC) analysis techniques for 50
short- period, single-planet systems discovered with radial velocity technique.
We develop a new method for accessing the significance of a non-zero orbital
eccentricity, namely {\Gamma} analysis, which combines frequentist bootstrap
approach with Bayesian analysis of each simulated data set. We find the
eccentricity estimations from {\Gamma} analysis are generally consistent with
results from both standard MCMC analysis and previous references. The {\Gamma}
method is particular useful for assessing the significance of small
eccentricities. Our results suggest that the current sample size is
insufficient to draw robust conclusions about the roles of tidal interaction
and perturbations in shaping the eccentricity distribution of short-period
single-planet systems. We use a Bayesian population analysis to show that a
mixture of analytical distributions is a good approximation of the underlying
eccentricity distribution. For short-period planets, we find the most probable
values of parameters in the analytical functions given the observed
eccentricities. These analytical functions can be used in theoretical
investigations or as priors for the eccentricity distribution when analyzing
short-period planets. As the measurement precision improves and sample size
increases, the method can be applied to more complex parametrizations for the
underlying distribution of eccentricity for extrasolar planetary systems.
