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 ...

3. Jakob Bernoulli and the Law of Large NumbersI consider Bernoulli’s main work, the Ars conjectandi (AC), which blazeda new trail by proving that statistical probability can be considered on a parwith the theoretical probability. Also described is the work of hiscontemporaries.Key words: law of large numbers, statistical probability, moral certainty,stochastic arguments3.1. Bernoulli’s Works3.1.1. The Diary (Meditationes). There, Bernoulli studied games ofchance and the stochastic side of civil law. He (1975, p. 47) noted that theprobability of a visitation of a plague in a given year was equal to the ratioof the number of these visitations during a long period of time to the numberof years in that period. He thus applied the definition of probability of anevent (of statistical probability!) rather than making use of chances. On p.46, in a marginal note, he wrote out the imprint of a review of Graunt’s book(§ 2.1.4) which Bernoulli possibly had not seen. But the most important inthe Meditationes is a (fragmentary) proof of the LLN which means thatBernoulli proved it not later than in 1690.3.1.2. The Art of Conjecturing (1713). Its Contents. Niklaus Bernoullicompiled a Preface (J. Bernoulli 1975, p. 108) where, for the first time ever,the term calculus of probability (in Latin) had appeared. The book itselfcontained four parts. Interesting problems are solved in parts 1 and 3 of theAC (the study of random sums for the uniform and the binomialdistributions, a similar investigation of the sum of a random number ofterms for a particular discrete distribution, a derivation of the distribution ofthe first order statistic for the discrete uniform distribution and thecalculation of probabilities appearing in sampling without replacement). Theauthor’s analytical methods included combinatorial analysis and calculationof expectations of winning in each set of finite and infinite games and theirsubsequent summing.Part 1 is a reprint of Huygens’ tract (§ 2.2.2). Bernoulli also compiled atable which enabled him to calculate the coefficients of x m in thedevelopment of (x + x 2 + … + x 6 ) n for small values of n. That polynomial tothe power of n was the generating function of the binomial (p + qx) with p =q.Part 2 dealt with combinatorial analysis and introduced the Bernoullinumbers.Part 4 contained the LLN. There also is an informal classical definition ofprobability (which Bernoulli had not applied when formulating that law), areasoning on the aims of the art of conjecturing (determination ofprobabilities for choosing the best solutions of problems, apparently in civillife) and elements of stochastic logic.Bernoulli likely considered the art of conjecturing as a mathematicaldiscipline based on probability as a measure of certainty and on expectationand including (the not yet formally introduced) addition and multiplicationtheorems and crowned by the LLN.In a letter of 3 Oct. 1703 Bernoulli (Kohli 1975b, p. 509) informedLeibniz about the progress in his work. He had been compiling it for many27

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