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

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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 ofred 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 theoryof 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

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