Limiting forms of the inverse beta probability density yield two distinct classes of distributions and functional forms for the entropy of the microcanonical ensemble. The two probability densities are the structure function and the probability distribution for maximum value. The structure function assures that the average value will also be the most probable value whereas the extreme-value distribution manifests a splitting where extreme values are over-whelmingly the most probable values. A thermodynamic criterion determining splitting in terms of the second derivatives of the two entropy expressions is given.