  We show how to analyze and interpret the correlation structures, the
conditional expectation values and correlation coefficients of exchangeable
Bernoulli random variables. We study implied default distributions for the
iTraxx-CJ tranches and some popular probabilistic models, including the
Gaussian copula model, Beta binomial distribution model and long-range Ising
model. We interpret the differences in their profiles in terms of the
correlation structures. The implied default distribution has singular
correlation structures, reflecting the credit market implications. We point out
two possible origins of the singular behavior.
