Example of Probability Distribution Not in Exponential Family

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SUMMARY

The discussion focuses on identifying a probability distribution whose support set is independent of its parameters and does not belong to the exponential family. Gaussian distributions, which have a support set of all real numbers, and Bernoulli distributions, defined by a parameter p, are highlighted as examples of distributions with parameter-dependent support. However, the participants express confusion regarding the prevalence of such distributions outside the exponential family. The conversation emphasizes the need for clear examples that meet these criteria.

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mynameisfunk
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Is there a good example of a probability distribution where the support set does not depend on the parameters and is still not a member of the exponential family?
 
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I'm a bit confused because it seems like almost every family of probability distributions will satisfy this. For example Gaussian random variables have a mean and a standard deviation, and their support set is all of \mathbb{R}. On the discrete case you have things like Bernoulli random variables with a parameter p that take value 1 with probability p, and -1 with probability 1-p.
 

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