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

More documents

Recommendations

Info

Notes1. Alimov [iv, § 2.1] introduced extended series of observations. O. S.2. Without saying anything else, I note that Tutubalin himself applied that methodin a joint paper (Tutubalin et al 2009, p. 189). O. S.3. That the electron is inexhaustible is Lenin’s celebrated statement from hisMaterialism and Empirical Criticism (1909, in Russian). The notion of electron isintrinsically contradictory, so perhaps the author indirectly stated the same aboutthose tables. Anyway, Lenin’s statement remains unjustified. O. S.BibliographyAlimov Yu. I. (1978 Russian), On the problem of applying the theory ofprobability considered by V. N. Tutubalin. Avtomatika, No. 1, pp. 71 – 82.Enin T. K. (1939 Russian), The results of an analysis of the assortment of hybridsof tomatoes. Doklady Akademii Nauk SSSR, vol. 24, No. 2, pp. 176 – 178. Alsopublished at about the same time in a foreign language in C. r. (Doklady) Acad. Sci.URSS.Ermolaeva N. I. (1939 Russian), Once more about the “pea laws”. Jarovizatsia,No. 2, pp. 79 – 86.Kolmogorov A. N. (1940), On a new confirmation of Mendel’s laws. C. r.(Doklady) Acad. Sci. URSS, vol. 28, No. 9, pp. 834 – 838.Tutubalin V. N. (1972), Teoria Veroiatnostei (Theory of probability). Moscow.Tutubalin V. N, Barabasheva Yu. M., Devyatkova G. N., Uger E. G. (2009Russian), Kolmogorov’s criteria and verification of Mendel’s heredity laws. Istoriko-Matematich. Issledovania, ser. 2, issue 13/48, pp. 185 – 197.138
VIOscar SheyninOn the Bernoulli Law of Large NumbersBernoulli considered (independent) trials with a constant probabilityof success, and rigorously proved that the frequency of success tendsto that probability. Mises, however, treated collectives, totalities ofphenomena or events differing from each other in some indication, andcharacterized by the existence of the limiting frequency of success andby irregularity. The latter property meant that for any part of thecollective that limiting frequency was the same.Alimov noted that artificially constructed collectives proved that theempirical frequency of success can become more stable as the numberof trials increased, but have no limit. Therefore, the existence of thatlimit is an experimental fact. I have described his viewpoint in somedetail in an Introduction to [v]. Tutubalin largely sided with Alimov.In the same Ars Conjectandi, previous to proving the LLN,Bernoulli stated that his law was also valid in its inverse sense (and DeMoivre independently stated the same with respect to the first versionof the CLT proved by him in 1733). In other words, an unknown andeven a non-existing probability (one of Bernoulli’s examples) could beestimated by the limiting frequency.In a little known companion paper (1765) to his main memoir(1764), Bayes all but proved his own limit theorem explicating thatinverse LLN. He did not make the final step from the case of a largefinite number of trials because he opposed the application of divergentseries which was usual in those times. That was done in 1908 byTimerding, the Editor of the German translation of Bayes, certainlywithout using divergent series.Bayes – Timerding examined the behaviour of the centred andnormed random variable η, the unknown probability, (η − Eη)/var ηwhereas the direct LLN dealt with the frequency ξ, (ξ − Eξ)/varξ. Hismain memoir became widely known and for a long time the Bayesapproach had been fiercely opposed, partly because an unknownconstant was treated as a random variable (with a uniformdistribution). Note that varη > varξ which is quite natural sinceprobability is only unknown in the inverse case. For attaining the sameprecision the inverse case therefore demands more trials than the directlaw. Mises could have called Bayes his main predecessor; actually,however, he only described the work of the English mathematician,and inadequately at that. Bayes completed the first stage of thedevelopment of probability theory.Alimov’s viewpoint was largely correct since he considered anincomparably more general pattern than Bernoulli and thought aboutthe necessary checks, but he [iv] was too radical in denying importantparts of mathematical statistics as also too brave in altering the Misesapproach. To borrow an expression from Tutubalin [end of ii], heintroduced the Mises approach of a light-weighted type.139
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 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
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
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: obtaining a deviation from the theo
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?