Probability Proofs for Stirling (and More): The Ubiquitous Role of 2π.
The Stirling approximation formula for n! dates from 1730. Here we give new and instructive proofs of this and related approximation formulae via tools of probability and statistics. There are connections to the Central Limit Theorem and also to approximations of marginal distributions in Bayesian s...
| Publicado en: | American Statistician Vol. 80; no. 3; pp. 405 - 413 |
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| Autores principales: | , |
| Formato: | Artículo |
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Taylor & Francis Ltd
Aug2026
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| Acceso en línea: | Ver este registro en EBSCOhost |
| fields | @attributes: recordID: 1 pdfLink: plink: https://search.ebscohost.com/login.aspx?direct=true&db=ssf&AN=195622627&site=ehost-live header: @attributes: shortDbName: ssf uiTerm: 195622627 longDbName: Social Sciences Full Text (H.W. Wilson) uiTag: AN controlInfo: bkinfo: jinfo: jid: 00031305 STT jtl: American Statistician issn: 00031305 maglogo: Y pubinfo: dt: Aug2026 vid: 80 iid: 3 pid: 377 pub: Taylor & Francis Ltd artinfo: ui: 195622627 10.1080/00031305.2025.2603256 ppf: 405 ppct: 8 formats: tig: atl: Probability Proofs for Stirling (and More): The Ubiquitous Role of 2π. aug: au: Hjort, Nils Lid Stoltenberg, Emil Aas affil: Department of Mathematics, University of Oslo, Oslo, Norway su: Central limit theorem Mathematical proofs Bayesian analysis Median (Mathematics) Mathematical formulas History of mathematics Approximation error Distribution (Probability theory) sug: subj: Central limit theorem Mathematical proofs Bayesian analysis Median (Mathematics) Mathematical formulas History of mathematics Approximation error Distribution (Probability theory) keyword: Binomial Central Limit Theorem Gamma variables Irwin–Hall Laplace Poisson Wallis We do history Binomial Central Limit Theorem Gamma variables Irwin–Hall Laplace Poisson Wallis We do history ab: The Stirling approximation formula for n! dates from 1730. Here we give new and instructive proofs of this and related approximation formulae via tools of probability and statistics. There are connections to the Central Limit Theorem and also to approximations of marginal distributions in Bayesian setups, with arguments which can be worked through by Master and PhD level students (and above). Certain formulae emerge by working through particular instances, some independently verifiable but others perhaps not. A particular case yielding new formulae is that of summing independent uniforms, related to the Irwin–Hall distribution. Yet further proofs of the Stirling flow from examining aspects of limiting normality of the sample median of uniforms, and from these again we find a proof for the Wallis product formula for π . A section detailing historical aspects and development is included, from Wallis 1656 and de Moivre and Stirling 1730 to Laplace 1778, etc. pubtype: Academic Journal doctype: Article src: R language: English refInfo: copyright: @attributes: flag: N holdings: @attributes: islocal: N |
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