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

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number of events) the advantage of the axiomatic approach is that itdistinctly separates the solution of stochastic problems into two parts:1. Choice of the mathematical model of the phenomenon orexperiment.2. Calculation within its limits.We are thus following Descartes’ advice: separate each probleminto so many parts that they become solvable. The first part, that is, thechoice of the mathematical model, is undoubtedly more difficult, andthe difficulty, as stated above, lies in determining the probabilities ofthe elementary events. A formulation of more or less general rules forovercoming this difficulty demands an introduction of some newconcepts. We have considered the concept of classical probability;another useful concept is that of conditional probability, but it isexpedient to begin by considering usual operations on events. In theset-theoretic context now adopted these operations coincide with thosein the set theory.A sum (unification) of events. A sum A∪ B of events A and B is anevent consisting of those elementary events that enter into A or B (orboth).A product (intersection) of events. A product AB of events A and Bis an event consisting of those elementary events that enter both A andB.A complementary (contrary) event A of event A is an eventcomposed of those elements that do not enter event A.If an experiment concludes by one of those elementary outcomeswhich enter some event C, we say that event C had occurred. Thus, thesum of events A and B occurs if at least one of those events hasoccurred. The product AB occurs if both events A and B has occurred.Complement A of event A occurs if A has not occurred.Mathematically, conditional probability P(A/B) that A occurs if Bhas occurred is determined by the equalityP( AB)P( A / B) = , P( B) ≠ 0.P( B)It follows that P(AB) = P(B) P(A/B).The part played by the concept of conditional probability is revealedby its frequentist interpretation. Consider n experiments with events Aand B occurring or not in each and let µ A , µ B , and µ AB be the number ofoccurrences of events A, B, and AB. It is evident that µ AB is also thenumber of occurrences of event A in those experiments, and the ratioµ AB /µ B −the conditional frequency of event A if event B has occurred.ThenµABµ / ( )( / )µ = ABn P ABP A Bµ / n≈ P( B)=BBand the conditional probability is interpreted as the conditionalfrequency.16
Let the space of elementary events Ω be separated into parts B 1 , B 2 ,..., B n , so that Ω = B1 ∪ B2 ∪... ∪ Bnand no two sets B i and B j havecommon elements. Then, for any A ⊆ Ω we will haveA = AB1 ∪ AB2 ∪... ∪ ABnwhich means that the elementary events included in A are separatedinto those entering B 1 , B 2 , ..., B n and obviouslyP(A) = P(AB 1 ) + P(AB 2 ) + ... + P(AB n ).This follows from the definition of P(A), see § 2.1.By definition of conditional probabilitynP( A) = P( AB ) = P( B ) P( AB )n∑ ∑ (2.2)i i ii= 1 i=1which is the formula of complete probability.There is another, the so-called Bayes formulaP( ABi ) P( Bi ) P( A / Bi)P( Bi/ A) = =.nP( A)P( B ) P( A / B )∑i=1ii(2.3)We have derived formulas (2.2) and (2.3) by issuing from thedefinition of conditional probability and applying really trivialtransformations. They can not therefore be called substantialmathematical theorems, but they nevertheless play an important part.Let us first consider the application of formula (2.2). Suppose, forthe sake of definiteness, that event A means that some article isdefective and assume also that that event is not by itself statisticallystable; more definitely, that there are mutually exclusive conditions ofmanufacturing B 1 , B 2 , ..., B n such that given B i , it is possible toconsider P(A/B i ) so that statistical stability is present.Suppose now that all the products manufactured under thoseconditions are stored without being sorted out but that their sharecorresponding to condition B i is given and equal to P(B i ). Considernow an experiment in which one article is chosen at random andchecked. Two outcomes are possible: A (defective) and A (qualitysufficient). Its random extraction means that such an experiment isstatistically stable, P(A) is expressed by formula (2.2) andP( A) = 1 − P( A).Unjustified hope had been previously connected with the Bayesformula (2.3) since subjective interpretation of probability was notruled out. For example, when having hypotheses B 1 , B 2 , ..., B n trustedwith probabilities P(B 1 ), P(B 2 ), ..., P(B n ), it was thought that anexperiment was desirable for indicating the proper hypothesis.17
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: chances to draw a more suitable apa
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 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
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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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