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Let me start, that i know that it's not very difficult to generate a probability distribution. If one takes any positive integrable function and normalizes it, this results in a probability density. Furthermore, i am familiar with the usage of certain distributions, so an answer should not direct to "use this distribution for that problem".

This question to the history of the normal distribution states, that

the first derivations must have come as a byproduct from trying to find fast ways to compute basic discrete probability distributions, such as binomials.

But how did things go on and lead us to the "discovery" of distributions like Cauchy, Gumbel, Weibull, ...

This wikipedia article provides an overview on the relationship among probability distributions. I wonder, how all of them were discovered and in fact got their legitimacy for applying in statistical practice!

Were there some "real world" problems leading to the discovery of well-known statistical distributions? Were they developed as special cases of other distributions and later become useful on certain problems? I know, sounds like a chicken-egg-problem, but how was the typical process in discovering new distributions in average?

I appreciate any hint on problems or applications, leading to the discovery of a certain distribution. Also references on the history of their development would be very useful.

The question:

How were statistical distributions discovered?

Edit:

In particular, i am interested in the typical process leading to the discovery of statistical distributions. Rather than going through the history of a certain distribution, i would like to understand the general mechanisms who resulted in "new" distributions.

skoestlmeier
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    You do not mean the discovery of the *concept* of statistical distributions but instead the discovery of any particular distribution? Isn't this a bit broad? It is a bit like 'how are stackexchange questions made-up?' sometimes there is a practical issue other times there are theoretic ponderings. – Sextus Empiricus Aug 10 '18 at 14:33
  • It's a wonderful question about a fascinating topic, but I have to agree (reluctantly) with @Martijn that it's too broad for this site. Since we do curate threads that ask for lists of references, perhaps we can keep this thread open by interpreting it as a similar "community wiki" list. – whuber Aug 10 '18 at 14:50
  • Thanks @MartijnWeterings for your helpful comment. I respect the scope of CV exchange and regret if my question may be too broad. I refer to your comments in an additional "edit"-section in the question and hope this clarifies the scope of my question. – skoestlmeier Aug 10 '18 at 15:15
  • A lot of these distributions come from playing around with existing ones. The Gumbal naturally arises as the distribution of the maximum of an ever-increasing sequence of exponential distributions, pertinent when analyzing the probability of independent failures. The Cauchy is the result of the ratio of two independent standard normal distributions. – Alex R. Aug 10 '18 at 20:35
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    +1 despite being broad, because I'd like to read even a partial answer and it's refreshingly different from other questions asked. I've seem this been recommended a bit (https://www.amazon.com/History-Statistics-Measurement-Uncertainty-before/dp/067440341X). Has anyone read it? Does it cover the discovery of a few important distributions? @whuber – Mark White Aug 11 '18 at 02:53
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    One issue with this question is that a number of questions on site address some history for a number of specific distributions (I think we can count the normal, gamma and beta, for example). Is it suitable to subsume perfectly good questions that are already here with a larger one that's recognized as too broad? Would it be better to come up with a short list of distributions of interest and ask a question about each one that doesn't already have a good answer? – Glen_b Aug 11 '18 at 10:35
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    @Mark I can warmly recommend Stigler's books on the history of statistics. I reread them every decade or so. They don't focus on distributions *per se,* but they do cover important parts of the history of the Normal distribution and (I think) at least make passing reference to the use of the Poisson. Most distributions IMHO are not "discovered:" their mathematical forms were already known in most cases to Newton or Euler and were later *applied* to certain statistical problems. – whuber Aug 11 '18 at 16:56
  • I just got informed, that this question is being discussed on [reddit r/statistics](https://www.reddit.com/r/statistics/comments/9a6w7j/question_how_were_statistical_distributions/) with some interesting remarks! – skoestlmeier Aug 28 '18 at 15:20
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    Similar question about beta distr: https://stats.stackexchange.com/questions/186562/how-did-statistician-came-up-with-distribution-at-the-first-place – kjetil b halvorsen Mar 14 '19 at 15:58

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