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1 Oscar Sheynin Theory of Probability and Statistics As Exemplified in Short Dictums Second Revised and Enlarged Edition Berlin Oscar Sheynin Theory of Probability and Statistics As Exemplified in Short Dictums Second revised and enlarged edition Berlin, 2009 @ Oscar Sheynin 2009 www.sheynin.de ISBN 3-938417-98-6 1 Contents 1. Introduction 2. Theory of Probability, NNo. 1 – 214 2.1.Origin and Aims, NNo. 1 – 44 2.2. Main Notions, NNo. 45 – 99 2.3. Logical Difficulties and Mistaken Opinions, NNo. 100 –114 2.4. Moral Expectation, NNo. 115 – 121 2.5. Law of Large Numbers and Central Limit Theorem, NNo. 122 – 181 2.6. The Axiomatic Approach and the Frequentist Theory, NNo. 182 – 214 3. Statistics and Mathematical Statistics, NNo. 215 – 622 3.1. Origin and Aims, NNo. 215 – 271 3.2. Population and Moral Statistics, Insurance, Games of Chance, NNo. 272 – 305 3.3. Medical Statistics, NNo. 306 – 340 3.4. Administration of Justice, NNo. 341 – 358 3.5. Quetelet (apart from His Appearance Elsewhere), NNo. 359 – 373 3.6. Statistics and Mathematics, NNo. 374 – 402 3.7. Application in Science (General Considerations), NNo. 403 – 460 3.8. Application in Natural Science, NNo. 461 – 498 3.9. The Two Currents, NNo. 499 – 577 3.10. The Soviet Cul-de-Sac, NNo. 578 – 616 3.11. Econometry, NNo. 617 – 622 4. Mathematical Treatment of Observations, NNo. 623 – 755 4.1. Ancient Astronomers, NNo. 623 – 640 4.2. Seventeenth and Eighteenth Centuries, NNo. 641 – 689 4.3. The True Value of a Measured Constant and the Method of Least Squares, NNo. 690 – 751 4.4.The Determinate Branch of the Theory of Errors, NNo. 752 – 755 5. Randomness, NNo. 756 – 832 5.1. Previous Attitudes, NNo. 756 – 811 5.2. New Ideas. Modernity, NNo. 812 – 832 Bibliography Index of Names 2 Pour prévoir des mathématiques la vraie méthode est d’étudier leur histoire et leur état present. Poincaré (1909, p. 167) I do feel how wrongful it was to work for so many years at statistics and neglect its history. K. Pearson (1978, p. 1) 1. Introduction I am presenting a first-ever scientific collection of short sayings on probability and statistics expressed by most various men of science, many classics included, from antiquity to Kepler to our time. Quite understandably, the reader will find here no mathematical formulas and in some instances he will miss a worthy subject. Markov chains provide a good example: their inventor had not said anything about them suitable for my goal. Nevertheless, the scope of the collected dictums is amazingly broad which reflects both the discussions concerning the lack of a solid foundation of probability theory until the 1930s and the great extent of applications of probability and statistics. And I have also included two related and most important topics, the treatment of observations and randomness. The present arrangement of the collected material is the result of several attempts, but some further improvement might have perhaps been possible here. In any case, I have discovered that a chronological arrangement, even of each separate subject, is not good enough. Having been unable to check several sayings included in a mostly popular collection Gaither et al (1996), I copied them with proper references. I am reasonably sure that this work will prove useful both for scientific and pedagogic purposes, – for providing a background even in unexpected cases and for greatly stimulating the study of probability and statistics. As to a general description of the history of these disciplines, I refer readers to Hald (1990; 1998) and Sheynin (2005/2009), also available at www. sheynin.de This second edition is much extended, several mistakes have been corrected and the material is more properly arranged. 3 2. Theory of Probability 2.1. Origin and Aims 1. It is in basic attitudes towards the phenomenal world, in religious and moral teachings and barriers, that I incline to seek for an explanation of the delay [of the origin of probability theory]. Mathematics never leads thought, but only expresses it. Kendall (1956, § 31). Comment. In particular, Kendall mentioned the gamblers’ psychology. 2. Two persons claim a cloak [or aught else. The first says] The whole of it is mine. [The second says] Half of it is mine. [They share it in the ratio 3:1.] Mishnah , Baba Metria 1 1. Comment. A thousand years later Pascal would have given the same answer and stochastically justified it. Mishnah (Blackman 1951 – 1955) was compiled in the 4 th century of our era. It consists of more than 60 treatises and is the first part of the Talmud. 3. Neither joint-stock societies, nor banks, nor stock exchanges stood in need of probability; their demands on probability only appeared in the 19 th century, when methods of scientific gain superseded downright robbery. Mrochek (1934, p. 50). Comment. Mrochek could have also mentioned marine and life insurance, cf. Ostrogradsky (1847): “Do not pay more; … try to pay less so as to gain something. Do not worry about the insurance society: it will not incur losses”. 4. Joignant la rigueur des demonstrations de la science [matheseos] à l’incertitude du hasard, et coinciliant ces choses en apparence contraire, elle peut, tirant son nom des deux, s’arroger à bon droit ce titre stupéfiant: La Géomètrie du hasard. Pascal (1654/1998, p. 172). 5. Toutefois je veux croire qu’en considérant ces choses [games of chance] plus attentivement, le lecteur apercevra bientôt qu’il ne s’agit pas ici d’un simple jeu d’esprit, mais qu’on y jette les fondements d’une spéculation fort intéressante et profonde. Huygens (1657/1920, p. 58). 6. Ich wünschte, dass jemand verschiedenen Arten von Glücksspielen (bei denen es schöne Beispiele gibt für diese Theorie) mathematisch bearbeitete. Das wäre zugleich anziehend und nützlich und nicht unwürdig Deiner oder eines anderen grossen Mathematikers. Leibniz, letter of 1703 to Jakob Bernoulli, in Latin; Gini (1946, p. 404). 7. There is a given Number of each of several Sorts of Things … put promiscuously together; out of which a given Number … is to be taken as it happens; To find the probability that there shall come out precisely a given Number of each Sort … T. Simpson (1740, Problem 6). Comment. Ostrogradsky (1848) considered an equivalent problem which he reasonably thought to be useful for, as we say now, statistical control of supplies, but his main formula (p. 342) was extremely involved and hardly ever checked by anyone else. My main point here is that Huygens (1657, Additional problem No. 4) 4 solved the same problem as Simpson did which goes to show that the emerging theory of probability was indeed not a “simple jeu d’esprit” (see No. 5). See also No. 8. And here is Poisson (1837, p. 1): Les géomètres du XVII e siècle qui se sont occupés du calcul des probabilités, ne l’ont employé qu’à déterminer les chances de différents jeux de cette époque; et ce n’est que dans le siècle suivant qu’il a pris toute son extension, et qu’il est devenu une des principales branches des mathématiques, soit par le nombre et l’utilité de ses applications, soit par le genre d’analyse auquel il a donné naissance. 8. An insect is placed on a table; find the chance that it will be found at time t at distance r from the point where it has started; if the insect be supposed to hop, we have the simpler case of which that of the crawling insect is the limit. Glaisher (1873). Comment. Random walks with applications to natural science had begun to be studied in earnest from the mid-19 th century (Dutka 1985), but games of chance can be interpreted as such. 9. The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc’d to a Mathematical Reasoning, and when they cannot, it’s a Sign our Knowledge of them is very small and confus’d. And where mathematical reasoning can be had, it’s a great folly to make use of any other, as to grope for a thing in the dark, when you have a Candle standing by you. I believe the Calculation of the Quantity of probability might be improved to a very useful and pleasant Speculation, and applied to a great many Events which are accidental, besides those of Games … Arbuthnot (1692, Introduction) as quoted by Gaither et al (1996). Comment. Todhunter (1865, p. 49) believed that that work was a translation of the Huygens tract (1657). 10. We should not take into account the number of the doubts but rather consider how great is their incongruity and what is their disagreement with what exists. Sometimes a single doubt is more powerful than a thousand other doubts. Maimonides (1963, II-23). Comment. Cf. J. Bernoulli (1713, pt. 4, chapter 3 and beginning of chapter 2). In chapter 2 he stated: “Probabilities are estimated both by the number and the weight of the arguments …” 11. To make conjectures about something is the same as to measure its probability. Therefore, the art of conjecturing or stochastics ( ars conjectandi sive stochastice ) is defined as the art of measuring the probability of things as exactly as possible, to be able always to choose what will be found the best, the more satisfactory, serene and reasonable for our judgements and actions. This alone supports all the wisdom of the philosopher and the prudence of the politician. Jakob Bernoulli (1713/2005, p. 17). 12. Under uncertain and dubious circumstances we ought to suspend our actions until more light is thrown.
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