# Is there just one exponential family? Or are there many exponential families? [duplicate]

I'm confused by the phrasing I've seen about exponential families. What does it mean to say "an" exponential family. Why not "the" exponential family?

From a pdf from Berkely: "we define an exponential family of probability distributions as those distributions..."

From Statistical Inference (George Casella, Roger L. Berger): "Let $$X_1, X_2, ..., X_n$$ be iid observations from a pdf $$f(x|\theta)$$ that belongs to an exponential family..."

I've also ready things that say, "this distribution belongs to an exponential family..."

But what are the different exponential families? Why not just say the exponential family? If there are multiple exponential families, why haven't I ever seen something like, "the binomial distribution belongs to exponential family A, while this other distribution belongs to family B..."?

I've searched around, and can't find a list of these families. How can there be an exponential family if there are not more than one of them?

• In these contexts, an individual exponential family is something like a normal, Poisson, gamma, etc. I was initially a bit confused about this as well... Mar 2, 2020 at 17:56
• "Exponential family" is like "sports car", a specific class with many specific examples, yet clear distinctions from others. Mar 2, 2020 at 18:22