  The stochastic gravitational wave background for pulsar timing arrays is
often modeled by a Gaussian ensemble which is isotropic and unpolarized.
However, the Universe has a discrete set of polarized gravitational wave
sources at specific sky locations. Can we trust that the Gaussian ensemble is
an accurate description? To investigate this, we explicitly construct an
ensemble containing $N$ individual binary sources with circular orbits. The
orbital inclination angles are randomly distributed, hence the individual
sources are elliptically polarized. We then compute the first two moments of
the Hellings and Downs correlation, as well as the pulsar-averaged correlation
mean and (cosmic) variance. The first moments are the same as for a previously
studied ensemble of circularly polarized sources. However, the second moments,
and hence the variances, are different for the two ensembles. While neither
discrete source model is exactly described by a Gaussian ensemble, we show that
in the limit of large $N$, the differences are small.
