  We investigate what can be learned about a population of distant KBOs by
studying the statistical properties of their light curves. Whereas others have
successfully inferred the properties of individual, highly variable KBOs, we
show that the fraction of KBOs with low amplitudes also provides fundamental
information about a population. Each light curve is primarily the result of two
factors: shape and orientation. We consider contact binaries and ellipsoidal
shapes, with and without flattening. After developing the mathematical
framework, we apply it to the existing body of KBO light curve data. Principal
conclusions are as follows. (1) When using absolute magnitude H as a proxy for
size, it is more accurate to use the maximum of the light curve rather than the
mean. (2) Previous investigators have noted that smaller KBOs have
higher-amplitude light curves, and have interpreted this as evidence that they
are systematically more irregular in shape than larger KBOs; we show that a
population of flattened bodies with uniform proportions could also explain this
result. (3) Our analysis indicates that prior assessments of the fraction of
contact binaries in the Kuiper Belt may be artificially low. (4) The pole
orientations of some KBOs can be inferred from observed changes in their light
curves; however, these KBOs constitute a biased sample, whose pole orientations
are not representative of the population overall. (5) Although surface
topography, albedo patterns, limb darkening, and other surface properties can
affect individual light curves, they do not have a strong influence on the
statistics overall. (6) Photometry from the OSSOS survey is incompatible with
previous results and its statistical properties defy easy interpretation. We
also discuss the promise of this approach for the analysis of future, much
larger data sets such as the one anticipated from the Rubin Observatory.
