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

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IV. N. TutubalinTheory of Probability in Natural ScienceTeoria Veroiatnostei v Estestvoznanii. Moscow, 1972IntroductionEven from the time of Laplace, Gauss and Poisson the theory ofprobability is using a complicated mathematical arsenal. At present, itis applying practically the entire mathematical analysis including thetheory of partial differential equations and in addition, beginning withKolmogorov’s classic (1933), measure theory and functional analysis.Nevertheless, books on the theory of probability for a wide circle ofreaders usually begin by stating that the fundamental problems ofapplying it are quite simple for a layman to understand. That wasCournot’s (1843) opinion, and we wish to repeat his statement righthere.However, it could have been also stated that those problems aredifficult even for specialists since scientifically they are still not quiteclear. More precisely, when discussing fundamental stochasticproblems, a specialist fully mastering its mathematical tools has noadvantage over a layman since they do not help here. In this case,important is an experience of concrete applications which for amathematician is not easier (if not more difficult) to acquire than foran engineer or researcher engaged in direct applications.At present, ideas about the scope of the theory of probability tookshape a bit more perfectly than in the time of Laplace and Cournot. Webegin by describing them.1. Does Each Event Have Probability?1.1. The concept of statistical stability (of a statistical ensemble).Textbooks on the theory of probability, especially old ones, usuallystate that each random event has probability whereas a random event issuch that can either occur or not. Several examples are offered, such asthe occurrence of heads in a coin toss or of rain this evening or asuccessful passing of an examination by a student etc. As a result, thereader gets an impression that, if we do not know whether a givenevent happens or not, we may discuss its probability, and the theory ofprobability thus becomes a science of sciences, or at least anabsolutely special science in which some substantial inferences may bereached out of complete ignorance.Modern science naturally vigorously rejects that understanding ofthe concept of probability. In general, science prefers experimentswhose results are stable, i. e. such that the studied event invariablyoccurs or not. However, such complete stability of results is notalways achievable. Thus, according to the views nowadays accepted inphysics, it is impossible for experiments pertaining to quantummechanics. On the contrary, it can be considered established6
sufficiently securely that a careful and honest experimentalist can inmany cases achieve statistical, if not complete stability of his results.As it is now thought, events, connected with such experiments, areindeed comprising the scope of the theory of probability. And so, thepossibility of applying the theory of probability is not, generallyspeaking, presented for free, it is a prize for extensive and painstakingtechnical and theoretic work on stabilizing the conditions, andtherefore the results, of an experiment. But what exactly is meant bystatistical stability for which, as just stated, we ought to strive? How todetermine whether we have already achieved that desired situation, orshould we still perfect something?It should be recognized that nowadays we do not have an exhaustiveanswer. Mises (1928/1930) had formulated some pertinent demands.Let µ A be the number of occurrences of event A in n experiments, thenµ A /n is called the frequency of A. The first demand consisted in that thefrequency ought to become near to some number P(A) which is calledthe probability of the event A and Mises wrote it down aslim µ A /n = P (A), n → ∞.In such a form that demand can not be experimentally checked since itis practically impossible to compel n to tend to infinity.The second demand consisted in that, if we had agreed beforehandthat not all, but only a part of the trials will be considered (forexample, trials of even numbers), the frequency of A, calculatedaccordingly, should be close to the same number P (A); it is certainlypresumed that the number of trials is sufficiently large.Let us begin with the merit of the Mises formulation. Properlyspeaking, it consists in that some cases in which the application of thetheory of probability would have been mistaken, are excluded, andhere the second demand is especially typical; the first one is apparentlywell realized by all those applying the theory of probability and nomistakes are occurring here.Consider, for example, is it possible to discuss the probability of anarticle manufactured by a certain shop being defective 1 . One of thecauses of defects can be the not quite satisfactory condition of a part ofworkers, especially after a festive occasion. According to the secondMises demand, we ought to compare the frequency of defectivearticles manufactured during Mondays and the other days of the week,and the same applies to the end of a quarter, or year due to the rushwork. If these frequencies are noticeably different, it is useless todiscuss the probability of defective articles. Finally, defective articlescan appear because of possible low quality of raw materials, deviationfrom accepted technology, etc.Thus, knowing next to nothing about the theory of probability, andonly making use of the Mises rules, we see that for applying the theoryfor analyzing the quality of manufactured articles it is necessary tocreate beforehand sufficiently adjusted conditions. The theory ofprobability is something like butter for the porridge: first, you ought toprepare the porridge. However, it should be noted at once that thetheory of probability is often most advantageous not when it can be7
Page 1 and 2: Studies in the History of Statistic
Page 3 and 4: Introduction by CompilerI am presen
Page 5: (Lect. Notes Math., No. 1021, 1983,
Page 9 and 10: is energy?) from chapter 4 of Feynm
Page 11 and 12: demand to apply transfinite numbers
Page 13 and 14: for stating that Ω consists of ele
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Page 17 and 18: Let the space of elementary events
Page 19 and 20: 2.3. Independence. When desiring to
Page 21 and 22: Eξ = ∑ aipi.Our form of definiti
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Page 25 and 26: where x is any real number. If dens
Page 27 and 28: probability can be coupled with an
Page 29 and 30: Nowadays we are sure that no indepe
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Page 33 and 34: (1/B n )(m − A n )instead of the
Page 35 and 36: along with ξ. For example, if ξ i
Page 37 and 38: µ( − p0) ÷np0 (1 − p0)nhas an
Page 39 and 40: distribution of the maximal term |s
Page 41 and 42: ξ (ω) + ... + ξ (ω)n1n{ω :|
Page 43 and 44: P{max ξ(t) ≥ x} = 0.01, 0 ≤ t
Page 45 and 46: 1. This example and considerations
Page 47 and 48: IIV. N. TutubalinTreatment of Obser
Page 49 and 50: structure of statistical methods, d
Page 51 and 52: Suppose that we have adopted the pa
Page 53 and 54: and the variances are inversely pro
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is applied with P(t) being a polyno
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ut some mathematical tricks describ
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It is clear therefore that no speci
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of various groups of machines, and
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nnA(λ) x sin λ t, B(λ) = x cosλ
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of the mathematical model of the Br
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dF(λ) = f (λ) dλ, so that B( t
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usually very little of them. Indeed
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This is the celebrated model of aut
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applications of the theory of stoch
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achieved by differentiating because
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u(x 1 , x 2 , t 1 , t 2 ) = v(x 1 ,
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Reasoning based on common sense and
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answering that question is extremel
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IIIV. N. TutubalinThe Boundaries of
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periodograms. It occurred that work
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at point x = 1. However, preceding
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He concludes that since the action
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The verification of the truth of a
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In the purely scientific sense this
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ought to learn at once the simple t
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the material world science had inde
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values of (2.1) realized in the n e
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*several dozen. The totality µ ica
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Mendelian laws. It is not sufficien
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example, the problem of the objecti
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a linear function is not restricted
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258 - 82 - 176 cases or 68.5% of al
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The Framingham investigation indeed
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or, for discrete observations,IT(ω
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What objections can be made? First,
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eliability and queuing are known to
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Kolman E. (1939 Russian), Perversio
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measurement is provided. Recently,
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which means that sooner or later th
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The foundations of the Mises approa
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A rather subtle arsenal is develope
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4.3. General remarks on §§ 4.1 an
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BibliographyAlimov Yu. I. (1976, 19
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processes are now going on in the s
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obtaining a deviation from the theo
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VIOscar SheyninOn the Bernoulli Law
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1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

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