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

wholly dominates now the teaching of mathematical analysis and anumber of other mathematical disciplines) which sharply separates thepertinent contents into mathematical and applied parts.At the beginning of the century textbooks on the theory ofprobability had contained very many real examples of statistical data;in the new textbooks such examples are disappearing. A naturalprocess of demarcating teaching mathematical theory and applicationsis possibly going on. Indeed, had we wished to include applications ina textbook on mathematical analysis, we would have to expoundmechanics, physics, probability theory and much other material.It is a fact, however, that the applications of mathematical analysisnaturally find themselves in courses and textbooks on mechanics andphysics, but that the applications of the theory of probability, whiledisappearing from textbooks on mathematical sciences, are not yetbeing inserted elsewhere. It follows that the main methods of properwork with actual data and, in particular, of how to decide whethersome statistical premises are fulfilled or not, are not includedanywhere.I have therefore thought it appropriate to insert here a part of thesemethods. They are indeed constituting its, so to say, didactical part.All such methods are particular, and are described in a natural way byconcrete examples. However, the inclusion of a few such examples,that seemed to me important for one or another reason, pursues inaddition another and more general aim. I attempted to prove that, inspite of a possible logical groundlessness, a stochastic investigationcan provide a practically doubtless result. Confidence intervals, criteriaof significance and other statistical methods to which, in particular,Alimov objects, are serving in these examples perfectly well and allowus to make definite practical conclusions. But of course, realapplications of probability theory both at the time of Laplace andnowadays are of a particular and concrete type. As to my attitudetowards all-embracing global constructions, it is sufficiently expressedin Chapter 1.2.1. On a new confirmation of the Mendelian laws. We explicateKolmogorov’s paper (1940) directly connected with the discussion ofbiological problems which took place then 18 .At first, some simple theoretical information. Suppose thatsuccessive repetitions of an observed event constitute a genuinestatistical ensemble and its results are values of some random variableξ. The results of n experiments are traditionally denotedx 1 , ..., x n (2.1)(not ξ 1 ,..., ξ n ) and F n (x) is called the empirical distribution function:F xthe number of x < x among all x ,..., x= (2.2)ni1 nn( ) .This function changes by jumps of size 1/n at points (2.1); for thesake of simplicity we assume that among those numbers there are noequal to each other. That function therefore depends on the random100