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

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solve scientifically many not less important problems from weatherforecasting to prevention of the flu.Elsewhere Tolstoi compares natural sciences with pleasures, −games, riding, skating, etc, outings, − and concludes that enjoymentshould not impede the main business of life. In his time, scientistsapparently yet constituted such a thin layer of the population, that thegreat writer had no occasion to feel the labouring principle of sciences’nature 16 . Briefly, natural sciences constitute one of the many spheresof human activity with all the thus following shortcomings and merits.Consequently, for example the criticism of the theory of probability ofthe speculative kind (cf. § 1.2) can only pursue restrictive aims.Indeed, it logically shows that the premises for applying that theorycan not be verified. This, however, concerns the premises of anyscience; although the lack of logic undoubtedly somewhat lowers thecertainty of knowledge, in many cases the conclusions of probabilitytheory still have a quite sufficient certainty for admitting them asscientific.Many authors including Laplace discussed how the practicalapplicability and certainty of those conclusions is established. Hisreasoning in the Essai is not rich in content and is reduced to statingthat induction was not reliable [cf. his criticism of Bacon in § 1.1] andthat analogies were still worse. In my context, the response is utmostsimple: the practical verification is achieved by the work of manypeople and many generations; they ever again return to studying agiven problem.If several large boulders were lying on a peasant’s plot, he had tobypass them when ploughing. But if his son becomes able to removethem, he will do it. Just the same, in science it is not forbidden toapproach old problems by new methods and either to confirm or refutethe previous results. In statistics, this means that, having a smallamount of data, it is impossible to say anything in a certain way, butduring a prolonged statistical investigation, with new material beingever again available, no doubts are finally left.Alimov is in the right when asserting that, having one sample, it isnot at all possible to verify whether we are dealing with independentrandom variables. However, the situation is sharply changed after afew new samples become available. Then, in particular, we can checkthe previously calculated confidence intervals.I had occasion to encounter some people keeping to logicalreasoning for whom the very concept of statistical testing ofhypotheses caused a feeling of displeasure. That concept from the verybeginning fixes the level of significance, i. e. some non-zeroprobability to reject mistakenly an actually true hypothesis. Someconsider this unacceptable, but the process of cognition does notconsist of a single test, and even when we reject a hypothesis, we donot, happily, pass a death sentence. If new data appear, we will test itanew.Tolstoi would have hardly rejected the viewpoint that science issome sphere of labour not higher, not lower than any other sphere(industry, agriculture, fishing etc). To support this assumption I cancite his admission, in the same book, that in its sphere of cognition of98
the material world science had indeed essentially advanced. Andmodern development leaves no doubt in the existence of the really truescience in contrast to the false science.What are the practical conclusions from the considerations above?Once we acknowledge science as a kind of active human work, itfollows, on the one hand, that at each moment it is incomplete andfragmentary; indeed, active work always lacks something (or evenvery much). On the other hand, what also follows is universality: manwill always engage in science and attempt to widen the sphere of thecertainty known.In a number of fields of application of mathematics and probabilitytheory in particular to real phenomena the situation became abnormalsince the practical possibilities of application are overestimated. Insuch cases it is expedient to stress the unavoidable fragmentary state ofall the existing applications: in mathematics, too grand intentions canoccur unattainable and their inevitable failure will create for thatscience an extremely undesirable blow to its prestige, a situation inwhich science can not normally develop.Thus, some years ago it was thought that, had there occurred apossibility of solving great problems of linear programming coveringthe economics of the entire nation, economic planning should bereorganized on that foundation. It is now absolutely clear that such aproblem can not be either formulated or solved at least because, giventhat global setting, such a notion of linear programming as set ofpossible technological methods has no sense 17 . As a result, the study oflocal problems for which linear programming can be effective, is not atall sufficiently developed.Awkward and absolutely useless concepts emerge when attemptingto combine global problems of linear programming with a stochasticdescription of the possible indeterminateness. Here also only properlyisolated local problems can have sense. In general, when applying theprobability theory to describe an indeterminate situation, it isextremely important to attain some unity between the extent ofroughing out the reality still admissible for a stochastic model and theamount of information to be extracted from reality for determining theparameters of the model. This situation is perfectly well described bythe proverb: You can not run with the hare and hunt with the hounds.In other words, a model that adequately describes reality in detail candemand so much information for determining its parameters, that it isimpossible to collect it. And a rough model only demanding a littleamount of statistical information can be unsuited for describing reality.The main demand on a researcher who practically applies the theory ofprobability is indeed to be able to find a way out of these difficulties.2. Logical and Illogical Applications of the Theory of ProbabilityFive years ago I thought it expedient to explicate, in a popularbooklet, the elements of the mathematical arsenal of probabilitytheory. However, almost at the same time as that booklet hadappeared, a sufficient number of textbooks on the theory of probabilityhad been published with the mathematical aspect being described evenmore than completely. Then, a tradition begins to take shape (and99
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Studies in the History of Statistic
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Introduction by CompilerI am presen
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(Lect. Notes Math., No. 1021, 1983,
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sufficiently securely that a carefu
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is energy?) from chapter 4 of Feynm
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demand to apply transfinite numbers
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for stating that Ω consists of ele
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chances to draw a more suitable apa
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Let the space of elementary events
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2.3. Independence. When desiring to
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Eξ = ∑ aipi.Our form of definiti
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absolutely precisely if the pertine
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where x is any real number. If dens
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probability can be coupled with an
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Nowadays we are sure that no indepe
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λ = λ(T)with λ(T) being actually
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(1/B n )(m − A n )instead of the
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along with ξ. For example, if ξ i
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µ( − p0) ÷np0 (1 − p0)nhas an
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distribution of the maximal term |s
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ξ (ω) + ... + ξ (ω)n1n{ω :|
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P{max ξ(t) ≥ x} = 0.01, 0 ≤ t
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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
Page 55 and 56: It is interesting therefore to see
Page 57 and 58: is applied with P(t) being a polyno
Page 59 and 60: ut some mathematical tricks describ
Page 61 and 62: It is clear therefore that no speci
Page 63 and 64: of various groups of machines, and
Page 65 and 66: nnA(λ) x sin λ t, B(λ) = x cosλ
Page 67 and 68: of the mathematical model of the Br
Page 69 and 70: dF(λ) = f (λ) dλ, so that B( t
Page 71 and 72: usually very little of them. Indeed
Page 73 and 74: This is the celebrated model of aut
Page 75 and 76: applications of the theory of stoch
Page 77 and 78: achieved by differentiating because
Page 79 and 80: u(x 1 , x 2 , t 1 , t 2 ) = v(x 1 ,
Page 81 and 82: Reasoning based on common sense and
Page 83 and 84: answering that question is extremel
Page 85 and 86: IIIV. N. TutubalinThe Boundaries of
Page 87 and 88: periodograms. It occurred that work
Page 89 and 90: at point x = 1. However, preceding
Page 91 and 92: He concludes that since the action
Page 93 and 94: The verification of the truth of a
Page 95 and 96: In the purely scientific sense this
Page 97: ought to learn at once the simple t
Page 101 and 102: values of (2.1) realized in the n e
Page 103 and 104: *several dozen. The totality µ ica
Page 105 and 106: Mendelian laws. It is not sufficien
Page 107 and 108: example, the problem of the objecti
Page 109 and 110: a linear function is not restricted
Page 111 and 112: 258 - 82 - 176 cases or 68.5% of al
Page 113 and 114: The Framingham investigation indeed
Page 115 and 116: or, for discrete observations,IT(ω
Page 117 and 118: What objections can be made? First,
Page 119 and 120: eliability and queuing are known to
Page 121 and 122: Kolman E. (1939 Russian), Perversio
Page 123 and 124: measurement is provided. Recently,
Page 125 and 126: which means that sooner or later th
Page 127 and 128: The foundations of the Mises approa
Page 129 and 130: A rather subtle arsenal is develope
Page 131 and 132: 4.3. General remarks on §§ 4.1 an
Page 133 and 134: BibliographyAlimov Yu. I. (1976, 19
Page 135 and 136: processes are now going on in the s
Page 137 and 138: obtaining a deviation from the theo
Page 139 and 140: VIOscar SheyninOn the Bernoulli Law
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1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

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