  Starting with just the assumption of uniformly distributed orbital
orientations, we derive expressions for the distributions of the Keplerian
orbital elements as functions of arbitrary distributions of eccentricity and
semi-major axis. We present methods for finding the probability density
functions of the true anomaly, eccentric anomaly, orbital radius, and other
parameters used in describing direct planetary observations. We also
demonstrate the independence of the distribution of phase angle, which is
highly significant in the study of direct searches, and present examples
validating the derived expressions.
