1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

The verification of the truth of a scientific proposition by practice,in the first place concerning fundamental sciences, has a specialproperty, namely, that it often takes more than a generation.Consequently, at least because of this the transfer of interest in sciencefrom one generation to the next one is essentially important.On the other hand, it is also important to bring that generalpsychological arrangement in correspondence with the actual results.Such efforts are going on in all sciences under differing circumstances.In the theory of probability the tension of passions is somewhatstronger than, say, in mathematics as a whole: it is possibly partlyconnected with Laplace. He was at the source of modern probabilityand the literary merits of his contribution laid an excessive discrepancybetween its emotional and philosophical and its concrete scientificaspects.The too wide general hopes are characterized by the emotionalshortcoming of changing into disappointment once encountering a realproblem. In a purely scientific aspect it consists in that the researcher,when formulating new problems, is not sufficiently critical. As aresult, efforts and material values are spent on futile attempts to solveproblems whereas the impossibility of achieving this would be obvioushad he been a bit more critical.In any case, certain ideas were being developed in scienceconcerning the sphere of application of the stochastic methods.Actually, each scientist, who carried out some applied study involvingprobability theory, made a certain contribution to these ideas.However, their clear formulation (brilliant also in the purely literarysense) is due to Mises (1928, p. 14). He himself also attempted toconstruct a peculiar mathematical foundation of the theory ofprobability which stirred up animated criticism and at present thegenerally recognized axiomatization of probability is that provided byKolmogorov (1933/1974). Nevertheless the concept itself of practicalapplication largely follows Mises’ idea.I remind briefly this concept of statistical homogeneity or statisticalensemble (collective). For ascertaining the principles I restrict myattention to the most simple case when an experiment can either leadto the occurrence of some event A or not. Denote by n A the number ofits occurrences in n experiments repeated under presumably the sameconditions. The ratio n A /n is called the frequency of the occurrence ofevent A. Even before Mises statisticians (for example Poisson whostudied the probability of judicial verdicts) understood perfectly wellthat for the applicability of stochastic methods to study the event A thestability of the frequency n A /n as n increases should experience everless fluctuations and tend, in some sense, to a limit (which is indeedunderstood as the probability P(A) of A).Mises supplemented these ideas by a clear formulation of anotherproperty that was also intuitively perfectly well understood bystatisticians. Here it is. Separate the n trials beforehand intosufficiently large totalities n 1 , n 2 , ..., then the respective frequenciesn A /n 1 , n A /n 2 , ... should also be close to each other. The separation oughtto be done by drawing on the previous information; thus, two totalitiescould have been trials done in summer and winter with the frequencies93