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1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

atio at birth. Nevertheless, it was Bayes (§ 5.2) who investigated theinverse case.Poisson proved his LLN by issuing from the CLT which he (1837a,pp. 254 – 271) was yet unable to justify rigorously; he had not evenstated the imposed conditions clearly enough. He also applied the CLTfor estimating the significance **of** discrepancies between indicatorsobtained from different series **of** observations and Cournot (1843,Chapters 7 and 8) borrowed his findings without mentioning him.Poisson proved the CLT even earlier (1824; 1829). In the first instancehe introduced the Cauchy distribution and found out (1824, §§ 4 and6) that it was stable.Statisticians only recognized the LLN for the case **of** Bernoullitrials, and only when the probability **of** the studied event existed,otherwise they refused to turn to the theory **of** probability at all (§3.2.3). Even worse, as a rule, they only understood the LLN in a loosesense (Ibidem).8.2. Theory **of** ErrorsIn the theory **of** errors Poisson **of**fered his pro**of** **of** the CLT and adistribution-free test for the evenness **of** the density **of** observational errors(1829, § 10). When discussing the precision **of** firing, Poisson (1837b, p. 73)stated that the less was the scatter (the appropriate variance) **of** hit-points,the better was the gun. He thus made a step towards recognizing Gauss’choice **of** least variance as the criterion for adjusting observations (§ 9.1.3),but he followed Laplace and never mentioned the Gauss theory **of** errorspartly since French mathematicians had been reasonably angered by Gauss’attitude towards Legendre (§ 9.1.1). One **of** his problems (1837b, § 7)consisted in determining the distribution **of** the square **of** the distance **of**some point from the origin given the normal distributions **of** the point’sdistances from the two coordinate axes. He thus was perhaps the first to treatclearly the densities as purely mathematical objects.8.3. Criminal **Statistics**Unlike Laplace, Poisson introduced the prior probability **of** thedefendant’s guilt, not to be applied in individual cases. One **of**Poisson’s statements (1837a, pp. 375 – 376) is debatable: he thoughtthat the rate **of** conviction should increase with crime. At the sametime he (p. 21) recognized that criminality represented l’état moral denotre pays.The application **of** probability theory to jurisprudence had beencriticized time and time again. Poinsot (Poisson 1836, p. 380) called itune fausse application de la science mathématique and unwiselyquoted Laplace (1814/1886, p. XI) who had remarked that the theory**of** probability was very delicate. Unwisely, because the same Essaicontained a page (p. LXXVIII) entitled Application du calcul desprobabilités aux sciences morales where Laplace declared that suchapplications were the effets inévitables du progrès des lumières. Thesame Essai also contained three chapters devoted to such applicationsto say nothing **of** Laplace’s own work on criminal statistics.Then, Mill (1843/1886, p. 353) had called the application **of**probability to jurisprudence an opprobrium [disgrace] **of** mathematics.77