# 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... – WavesWashSands Mar 2 '20 at 17:56
• "Exponential family" is like "sports car", a specific class with many specific examples, yet clear distinctions from others. – AdamO Mar 2 '20 at 18:22
• We have many posts about this: see stats.stackexchange.com/search?q=%22exponential+family%22 – whuber Mar 2 '20 at 20:23
• Thanks for your responses. I think the missing piece for me was that a distribution itself could be a family. I just honestly didn't understand the idea of a "family" at all; but now I understand it's just a set of pdf's with a certain form with varying parameters. So binomial(n, p) is an example of an exponential family, and binomial(100, 0.5) is a member of that family? – Nick Koprowicz Mar 3 '20 at 1:49
• This shouldn't be closed in my opinion - it's a common point to get stuck on, and it's not exactly the topic any of the linked questions. (The accepted answer to the first linked question does answer it though.) – Nathaniel Mar 14 at 10:33