A formal definition of an exponential family is the starting point of a widely self-contained, mathematically rigorous and concise course on exponential families in a systematic structure. The definition is followed by a variety of detailed examples of one- and multiparameter, uni- and multivariate probability distributions. Basic notions and representations of exponential families, such as minimal and canonical representation, natural parametrization, and mean value parametrization are introduced and structural results are presented, which are illustrated by examples.
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