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Just for the sake of curiosity, I've been curious if the Gaussian distribution was derived from the Binomial, or if it was found by other means, and retrospectively associated with the Binomial. Anyway, I found this proof, which goes a bit over my head. The proof seems to center on Stirling's formula, which is an approximation technique for factorials, converging on ground truth values when limits are taken.

Could someone give me an intuitive explanation for (A) How/why Stirling's formula works? and (B) Why the Gaussian should seem as a natural extension from the Binomial?

jbuddy_13
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    I believe Abraham DeMoivre was the first to publish such a derivation c. 1733. A translation is available at https://www.york.ac.uk/depts/maths/histstat/demoivre.pdf. The methods are elementary -- there is no recourse to Stirling's formula. I gave an exposition of the basic idea in a discussion of the Central Limit Theorem at https://stats.stackexchange.com/a/3904/919. – whuber Apr 08 '21 at 17:23
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    A) Simply put: Stirling's formula approximates large combinatorial terms, which is exactly how the binomial distribution handles the ancillary arrangements of cases and non-cases in a sample of $n$. As whuber notes, this isn't necessary. B) The Gaussian isn't a "natural extension" from the binomial. In the case when $np \rightarrow c, c$ finite instead of $\infty$, the limiting distribution is Poisson, not normal. Ironically, while DeMoivre preceded Gauss, Gauss derived the eponymous distribution through other means. – AdamO Apr 08 '21 at 17:54
  • @AdamO, slight tangent addressing point A in your comment: Do programmatic implementations of the Binomial PMF evaluate the factorial terms using Stirling's formula? From context, it sounds that this is the case, but I don't want to walk away mistaken. – jbuddy_13 Apr 08 '21 at 19:30
  • https://www.r-project.org/doc/reports/CLoader-dbinom-2002.pdf – AdamO Apr 08 '21 at 19:45

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