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

probabilities **of** red and black coincide and Poincaré (p. 148) attempted tojustify that fact. Suppose that the needle stops after travelling a distance s(2π < s < A). Denote the corresponding density by φ(x), a functioncontinuous on [2π; A] and having a bounded derivative on the same interval.Then, as Poincaré demonstrated, the difference between the probabilities **of**red and black tended to zero as the length **of** each red (and black) arcbecame infinitesimal (or, which is the same, as s became infinitely large).He based his substantiation on the method **of** arbitrary functions (Khinchin1961/2004, pp. 421 – 422; von Plato 1983) and himself sketched its essence.c) Shuffling a deck **of** cards (p. 301). In an extremely involved manner,by applying hypercomplex numbers, Poincaré proved that after manyshuffling all the possible arrangements **of** the cards tended to becomeequally probable. See § 7.1-6.7) Mathematical treatment **of** observations. In a posthumously publishedRésumé **of** his work, Poincaré (1921/1983, p. 343) indicated that the theory**of** errors naturally was his main aim in the theory **of** probability. In histreatise he (pp. 169 – 173) derived the normal distribution **of** observationalerrors mainly following Gauss; then, like Bertrand, changed the derivationby assuming that not the most probable value **of** the estimator **of** the locationparameter coincided with the arithmetic mean, but its mean value. He (pp.186 – 187) also noted that, for small absolute errors x 1 , x 2 , …, x n , theequality **of** f (z) to the mean value **of** f(x i ), led to z, the estimate **of** the realvalue **of** the constant sought, being equal to the arithmetic mean **of** x i . Itseemed to him that he thus corroborated the Gauss postulate.Finally, Poincaré (p. 188) indicated that the variance **of** the arithmeticmean tended to zero with the increase in the number **of** observations andreferred to Gauss (who nevertheless had not stated anything at all about thecase **of** n → ∞). Nothing, however, followed since other linear means hadthe same property, as Markov (1899a/1951, p. 250) stated on anotheroccasion. Poincaré himself (1896/1912, pp. 196 – 201 and 217) twiceproved the consistency **of** the arithmetic mean. In the second case he issuedfrom a characteristic function **of** the type **of** (1) and (2) and passed on to thecharacteristic function **of** the arithmetic mean. He noted that, if that functioncould not be represented as (2), the consistency **of** the arithmetic mean wasquestionable, and he illustrated that fact by the Cauchy distribution. Perhapsbecause **of** all this reasoning on the mean Poincaré (p. 188) declared thatGauss’ rejection **of** his first substantiation **of** the MLSq was assez étrangeand corroborated this conclusion by remarking that the choice **of** theparameter **of** location should not be made independently from thedistribution (which directly contradicted Gauss’ mature approach). In thesame context Poincaré (p. 171) argued that everyone believed that thenormal law was universal: experimentalists thought that that was amathematical fact and mathematicians believed that it was experimental.8) Randomness. Poincaré discussed randomness both in his treatise and inhis scientific-popular booklets. In § 1.1 I noted his statement about the linkbetween randomness and necessity. There also, is a description **of** chaoticprocesses, and two **of** his explanations **of** chance. Maxwell (§ 10.8.5-3)anticipated one **of** these, but did not mention chance.I would argue that Poincaré initiated modern studies **of** randomness. Forhim, the theory **of** probability remained an accessory subject, and his almosttotal failure to refer to his predecessors except Bertrand testifies that he was120