  We use statistical and topological quantities to test the COBE-DMR first year
sky maps against the hypothesis that the observed temperature fluctuations
reflect Gaussian initial density perturbations with random phases. Recent
papers discuss specific quantities as discriminators between Gaussian and
non-Gaussian behavior, but the treatment of instrumental noise on the data is
largely ignored. The presence of noise in the data biases many statistical
quantities in a manner dependent on both the noise properties and the unknown
CMB temperature field. Appropriate weighting schemes can minimize this effect,
but it cannot be completely eliminated. Analytic expressions are presented for
these biases, and Monte Carlo simulations used to assess the best strategy for
determining cosmologically interesting information from noisy data. The genus
is a robust discriminator that can be used to estimate the power law
quadrupole-normalized amplitude independently of the 2-point correlation
function. The genus of the DMR data are consistent with Gaussian initial
fluctuations with Q_rms = 15.7 +/- 2.2 - (6.6 +/- 0.3)(n - 1) uK where n is the
power law index. Fitting the rms temperature variations at various smoothing
angles gives Q_rms = 13.2 +/- 2.5 uK and n = 1.7 +0.3 -0.6. While consistent
with Gaussian fluctuations, the first year data are only sufficient to rule out
strongly non-Gaussian distributions of fluctuations.
