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

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Note also that we often are unable to repeat an experiment with agiven probability of success for a sufficiently large number of times sothat it is perhaps impossible to find out p i , which does not necessarilyprevent us from calculating p . The approximate value (4.1) thereforehas much more chances to find practical application than the exactformula for P{µ = m}.Then, the demand of independence of the individual trials can beweakened: it may be assumed that they have more than two outcomesbut such possibilities exceed the limits of this booklet.Those possibilities of weakening the conditions of the Poissontheorem without changing its conclusion lead to the Poissondistribution attaching probabilities (4.1) to values m = 0, 1, 2, ...becoming one of the most universal laws. Consider for example theproblem of the number of refusals during time T in cases ofcomplicated systems. Suppose a system consists of n elements and p i =p i (T) is the probability of a refusal of the i-th element (and that afterrefusal the damaged element is instantly replaced). The number ofrefusals is the number of successes in n trials with the i-th trial beingconnected with the i-th element and its success means a refusal of thatelement. A given element can experience more than one refusal and itsrefusal can somewhat influence the refusals of the other elements, − allthe same, if p i are sufficiently low, we expect the Poisson distributionwith the parameter λ = n p to describe the number of refusals.Deviations are only possible if the connections between the refusalsof different elements are strong. Given low values of p i it is natural toexpect a linear dependence of p i = p i (T) on T:p ( T ) = p′T.iTheniλ = λ(T) = λ′T (4.3)and the probability of m refusals will beP(λ′T )m!m−λ′T{µ = m} ≈ e .For the probability of failure-free work during time T, that is, for µ =0, we have−λ′P{µ = 0} ≈ e Twhich is the generally known exponent law for the time of failure-freework.The Poisson and the exponent laws therefore correspond to eachother. There occurs some harmonious correspondence that we mayhope to apply beneficially for solving practical problems.Our model does not allow for aging; to achieve that we ought toreplace the linear dependence (4.3) by a more complicated dependence30
λ = λ(T)with λ(T) being actually approximated by a function of a most simplekind, for example by a polynomial. Its coefficients can be obtained byone or another method, for instance by the method of least squares.However, the number of parameters necessary to be included will inthis case be larger than when aging is not allowed for and, accordingly,the model will enjoy less faith.In general, with or without allowing for aging, it is natural to applythe Poisson law for describing the number of failures, only itsparameter is determined in differing ways. The fit of the Poisson law,its agreement with the actual data should be checked by statisticaltests. If a good agreement is lacking, it will be likely more natural tosuspect the statistical homogeneity of the data rather then theapplicability of the Poisson law. Only after checking that out may wethink about choosing another distribution for describing the number offailures.None of this certainly applies to the case of strong ties between thefailure of different elements. For example, if the failure of one part ofthe system automatically leads to a failure of its other part, the totalnumber of failures will be doubled. In such cases, even if the numberof initial failures follows the Poisson law, the doubled figure will not,and it is more natural to apply here the normal law. And the Poissonlaw is certainly only applicable when a failure really is a rare event.An excellent set of other examples of the application of the Poissonlaw is to be found in Feller (1950) only the theory of rare excursions ofstochastic processes can possibly be added to it. Just as any rare event,the number of such excursions beyond a sufficiently high level obeysthe Poisson law.4.2. The Central Limit Theorem. The Poisson law is determinedby a single parameter λ. It is not difficult to show that λ is theexpectation of a random variable distributed according to that law.Here, we will consider an even more universal stochastic law, the socallednormal law determined by two parameters, expectation andvariance. It was discovered at about the same time by Gauss andLaplace who issued from absolutely different considerations.Gauss discovered that exactly in the case of a normal law ofdistribution of the observational errors it is most natural to choose thearithmetic mean of the individual measurements as the estimator of thereal value of the measured magnitude. Laplace’s starting point was hisdiscovery of an extremely powerful method of calculating thedistribution of a sum of random variables. Gauss’ ideas are importantfor treating the results of measurement and were further developed inmathematical statistics as the so-called method of maximumlikelihood. Laplace’s ideas concerned the properties of arithmeticoperations on a large number of random variables and actuallyconstitute the foundation of modern probability theory. In this booklet,devoted to the theory of probability rather then mathematical statistics,we adopt the Laplacean approach 5 .Consider arbitrary independent random variables31
Page 1 and 2: Studies in the History of Statistic
Page 3 and 4: Introduction by CompilerI am presen
Page 5 and 6: (Lect. Notes Math., No. 1021, 1983,
Page 7 and 8: sufficiently securely that a carefu
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
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: Nowadays we are sure that no indepe
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 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 ,
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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
Page 117 and 118:
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
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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