Take beta distribution as an example
Just want to understand the thinking behind it, not the exact history. What was the problem what was the end result and how did beta distribution helped. (I am sure beta distribution was not the end goal)
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The very short story: In the beginning there were discrete distributions: tossing a coin, rolling a die, etc. I guess it helped people calculate odds in games of chance. Then, when people realized there is more than integers, continouus distributions emerged through the usual limit process you find anywhere in analysis (... pick ever more and finer partitions). In this way the beta distribution entered statistics as probably all distributions do: someday anyone finds out it is convenient because it
(i) reproduces well some real world behaviour,
(ii) stems from a mathematical derivation (central-limit theorem, maximum entropy, etc.),
(iii) has some nice analytical or numerical properties (closed expression for moments, analytic calculation of maximum likelihood, posteriors, etc.) which give an edge over other distributions.
I guess usually it's a combination of (i),(ii) and (iii), and the list is probably not exhaustive.