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

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probabilities for those magnitudes can be derived. Introduce a randomvariablef k (ξ) = 1, if ξ i = k; 0, if not, k = 1, 2, 3, ...Since ξ i is the number of failures for the i-th machine, the number ofmachines that had k failures is equal to ∑f k (ξ i ), a sum of independentrandom variables. For most machines λ i is near zero, therefore, if k ≠0, the probabilityP{f k (ξ i ) = 1}is low, and the sum above roughly obeys the Poisson distribution. Itsparameter is derived fromkλiE[ ∑ fk (ξi)] = ∑ E fk (ξi), E fk (ξi) = P{ξ i= k}= ek!ii−λiwhere, provided the hypothesis of statistical homogeneity is valid, λ i iscalculated as stated above. A simple calculation (Belova et al 1965,1967) indicates that at different values of k the studied sums are closeto independent random variables.How does deviation from statistical homogeneity reveal itself?Some machines will have a higher breakdown rate and experience twoor more failures, other will deviate in the opposite sense and workfailure-freely. When statistical homogeneity is corrupted, the numberof machines with two or more failures will increase, and will decreasefor those with one failure.The treatment of actual data resulted in the following number ofmachines with 1, 2, 3 and 4 failures (line 1) as compared with thecorresponding expectations (line 2).1. 27 10 1 12. 29.6 5.7 1.5 0.44The number of machines with one failure decreased insignificantly butof those with two failures increased noticeably: for the Poisson lawwith parameter 5.7 the probability of 10 or more is 0.065. For k = 3and 4 the deviations were small.The only deviation worth discussing is that for machines having 2failures. However, we may consider it maximal for four independentdeviations and then its probability is 1 − (1 – 0.065) 4 ≈ 0.25 so that itsdeviation is not especially significant.Although the hypothesis of statistical homogeneity had passed arather rigid test with credit, some shadow of doubt is still cast on it.This seems to mean that for most machines the breakdown rate isroughly the same but that small groups of them it can stand out. Awide scatter would have led to an essentially more significant result ofthe test. [...]It follows that in general the derived fitting mean curve p 2 (t) can beapplied for an approximate calculation of the mean number of failures62
of various groups of machines, and this provides us a test forestimating the reliability of the insulation. Purely statistical methodscertainly do not concern the improvement of that reliability which is atechnological problem. But at least we may say whether the reliabilityof insulation had changed and in which direction or that it remained asit was previously. This is the practical significance of the work donewhich would not be so important had the comparatively high statisticalhomogeneity of the insulation not been established. [...]2.4. The naked eye study. We had assumed that smoothing bypolynomials is usually successful because the data for that treatment isselected beforehand by naked eye. It would have been improper to failto mention that physicists and engineers also often perform thesmoothing itself by naked eye without applying the method of leastsquares. And how do we decide that the smoothing in a given case wassuccessful? Perhaps because the curve derived by least squares passesexactly where it would have been if drawn without applying thatmethod?An experimental smoothing by naked eye of the broken line in § 2.2was carried out. Participants were mathematicians, workers at astatistical laboratory, and engineers. Each received a list of paper withonly that line shown [...]. The results achieved by an overwhelmingmajority were very good. Fifteen out of sixteen of those participatinghad almost completely drawn their curves between the two curves,p 1 (t) and p 3 (t) as shown on Fig. 2. [...]I. V. Girsanov, the chief of one of the sections of the statisticallaboratory, achieved the best result; he unfortunately perished in a latertourist mountain tour. [...] In general, the results of smoothing bynaked eye are quite comparable in precision with the method of leastsquares. Had we been only interested in curve p 2 (t), we could havewell drawn it without any calculations. However, a thorough statisticaltreatment demands an estimation of precision as well for which astatistical model and science in general are necessary.Thus, when estimating the probability of success in Bernoulli trials,we turn to frequencies, but for understanding how large can thedeviations of frequency from probability be, we should, first, considerthe trials independent (the statistical model) and second, apply the DeMoivre – Laplace theorem which (however done) is proven in acomplicated manner [in essence, by the former in 1733] and this isundoubtedly science.When smoothing a broken line by naked eye, we do not even haveto know the number of observations used for calculating its points [...]and anyway it is impossible to indicate the confidence region for thecurve sought. Here, we need all the science connected with the methodof least squares and still the almost complete coincidence of the areashown on Fig.2 with that between the curves p 1 (t) and p 3 (t) demands tobe somehow explained.Note, however, that for small values of t that first area is somewhatnarrower than the second one whereas that latter, as shown bycalculation, is 1.5 – 2 times narrower there than a thoroughlyconstructed confidence region with the usual confidence coefficient of0.70 – 0. 95. This means that the indefiniteness of the naked eye63
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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
Page 11 and 12: demand to apply transfinite numbers
Page 13 and 14: for stating that Ω consists of ele
Page 15 and 16: chances to draw a more suitable apa
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
Page 23 and 24: absolutely precisely if the pertine
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
Page 31 and 32: λ = λ(T)with λ(T) being actually
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
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: It is clear therefore that no speci
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 and 98: ought to learn at once the simple t
Page 99 and 100: the material world science had inde
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
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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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