  We stacked the X-ray data from the ROSAT All Sky Survey for over 4,000
clusters selected from the 2MASS catalog and divided into five richness
classes. We detected excess X-ray emission over background at the center of the
stacked images in all five richness bins. The interrelationships between the
mass, X-ray temperature and X-ray luminosity of the stacked clusters agree well
with those derived from catalogs of X-ray clusters. Poisson variance in the
number of galaxies occupying halos of a given mass leads to significant
differences between the average richness at fixed mass and the average mass at
fixed richness that we can model relatively easily using a simple model of the
halo occupation distribution. These statistical effects probably explain recent
results in which optically-selected clusters lie on the same X-ray
luminosity-temperature relations as local clusters but have lower optical
richnesses than observed for local clusters with the same X-ray properties.
When we further binned the clusters by redshift, we did not find significant
redshift-dependent biases in the sense that the X-ray luminosities for massive
clusters of fixed optical richness show little dependence on redshift beyond
that expected from the effects of Poisson fluctuations. Our results demonstrate
that stacking of RASS data from optically selected clusters can be a powerful
test for biases in cluster selection algorithms.
