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 ...

The very fact described by formulas (1) and (2) was, however, extremelyimportant for the development of probability and statistics; and, anyway,should we deny the importance of existence theorems? Bernoulli’s resultproved that, given a large number of observations, statistical probabilityprovided moral certainty and was therefore not worse than the theoreticalprobability. His main aim was to discover whether the limit (2) existed andwhether it was indeed unity rather than a lesser positive number. The latterwould have meant that induction (from the ν trials) was inferior todeduction.Stochastic reasoning was now justified beyond the province of games ofchance, at least for the Bernoulli trials. Strangely enough, statisticians for along time had not recognized this fact. Haushofer (1872, pp. 107 – 108)declared that statistics, since it was based on induction, had no intrinsicconnections with mathematics based on deduction. And Maciejewski (1911,p. 96) introduced a statistical law of large numbers instead of the Bernoulliproposition that allegedly impeded the development of statistics. His ownlaw qualitatively asserted that statistical indicators exhibited ever lesserfluctuations as the number of observations increased and his opinion likelyrepresented the prevailing attitude of statisticians. Bortkiewicz (1917, pp. 56– 57) thought that the LLN ought to denote a quite general fact,unconnected with any stochastic pattern, of a degree of stability of statisticalindicators under constant or slightly changing conditions and a large numberof trials. Even Romanovsky (1912, p. 22; 1924, pt 1, p. 15; 1961, p. 127)kept to a similar view.That elementary understanding of the LLN has its prehistory, see thestatements of De Witt (§ 2.1.3) and Halley (§ 2.1.4). Again, it was thoughtthat the number of successes in n Bernoulli trials with probability p wasapproximately equal to np. Cardano applied this formula in calculationsconnected with games of dice (Ore 1953/1963, pp. 152 – 154 and 196).In astronomy, the arithmetic mean became the universal estimator of theconstant sought (§ 1.7). Recall also (§ 2.1.3) the practice of buying annuitiesupon several young lives. Boscovich (1758, § 481) had somewhat vaguelymaintained that the sum (not the mean!) of random magnitudes decreasedwith an increase in the number of terms (Gower 1993, p. 272). Mycorrection also applies to the other statements above to which I am nowadding Kepler (Sheynin 1973c, p. 120). He remarked that the total weight ofa large number of metal money of the same coinage did not depend on theinaccuracy in the weight of the separate coins. Even Helmert (1905/1993, p.200) had to refute that mistake.3.2.4. Randomness and Necessity. Apparently not wishing to encroachupon theology, Bernoulli (beginning of Chapter 1) refused to discuss thenotion of randomness and subjectively described the contingent but at thebeginning of Chapter 4 explained randomness by the action of numerouscomplicated causes, cf. § 11.3. The last lines of his book stated that somekind of necessity was present even in random things. He referred to Platowho had indeed taught that after a countless number of centuries everythingwill return to its initial state at the moment of creation. In accordance withthat archaic notion of the Great Year, Bernoulli thus unjustifiablygeneralized the boundaries of his law.30

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