TITLE: non-exponential family probability distributions and their uses
QUESTION [6 upvotes]: I was actually trying to find information on non-exponential family probability distributions. So many of the distributions that we study in statistics are members of an exponential family of distributions. But this leads to the natural question of whether there exist sets of distributions that are not members of an exponential family? I could not find any references to these. 
Does anyone know of distributions that are not members of an exponential family? Also, what are the applications for these types of distributions? Where are they used?
Thanks.

REPLY [2 votes]: I assume you're referring to distributions with unbounded tails. That rules out uniform, beta and so forth. The exponential distributions are a good watermark in the distributions manifold, splitting it into 3 subspaces: exponential, subexponential and superexponential. Examples of each are:
Exponential: Normal
Superexponential:
Subexponential: Lévy, Cauchy, Student t, Pareto, Generalised Pareto, Weibull, Burr, Lognormal, Log-Cauchy, Log-Gamma
The latter, the fat-tailed distributions, are particularly useful to Extreme Value Theory and when applied to the measurement of low-frequency/high-impact risks, such as in operational risk.