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

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µ µ µ µn n n n1 2 1 2var( − ) = var( ) + var( ).1 2 1 2If the terms in the right side are known, we could have said bymeans of a table of the normal distribution whether the mentioneddifference can be explained by purely random causes or not. And forcalculating those variances (but not at all for applying the central limittheorem) we have to assume that the trials in both series areindependent, that is, that two series of Bernoulli trials were madewhereas the central limit theorem does not demand completeindependence. Then, if p 1 = p 2 = p (whose value is unknown),µ µ 1 1n n n n1 2var( ) + var( ) = p(1 − p)( + ).1 2 1 2It can be shown that the unknown p may be replaced here byµ + µˆp =n + n1 21 2and, assuming that the probabilities were identical, we see thatµ µ 1 1ˆ ˆn n n n1 2ξ = ( − ) ÷ p(1 − p)( + )1 2 1 2has an approximately standard normal distribution. Now, ξ can also becalculated only by issuing from numbers (µ 1 , n 1 ) and (µ 2 , n 2 ), i. e., bythe known results of experimenting. It is known that the absolutevalues of ξ exceeding 2 or 3 are unlikely. It follows that, whenobtaining values of that order, we should conclude that either thehypothesis p 1 = p 2 does not hold or, if it does, that an unlikely eventhad taken place.How to choose between these conclusions? Some authors think thatthe decision theory can allegedly numerically express the risk of eachpossible choice and thus help here. However, the risk there isexpressed by magnitudes which either are senseless or in any case willnever be known to the researcher. The theory based on a quantitativeexpression of risk is always useless except in studies of games ofchance 8 . Actually, the choice between the two mentioned decisions is acomplicated procedure; at present, it is impossible to study it withinthe limits of a mathematical theory.When solving such problems, we have to compare somehow theimportance of each possible solution should it occur wrong. Bothscientific and moral considerations denoted by the word conscienceare involved here. Approximately similar but somewhat simpler ischecking the hypothesis that the probability of success p in a givenseries of Bernoulli trials equals a given number p 0 . If it is also true fora series of n trials with µ successes, then36
µ( − p0) ÷np0 (1 − p0)nhas an approximate standard normal distribution. Exactly bycalculating that magnitude can the hypothesis of a male birth beingequal to 1/2, see § 3.1, be checked (and rejected).The arc sine transformation. I have just expounded the principles ofchecking the equality of probabilities in two series of Bernoulli trials.Now, I aim at indicating by an example the pertinent convenientmethods developed in mathematical statistics. Nothing new inprinciple is here involved, but the practical convenience is essential.As an example, I choose the so-called arc sine transformationdiscovered by the celebrated English statistician Fisher. His idea wasvery simple: we consider some function f(µ/n) instead of µ/n, of thefrequency of success itself. We haveµ µ µf ( ) = f [( − p) + p] = f ( p) + f ′( p)( − p) + ...n n nFor large values of n, that frequency is close to p, so that we ignorethe other terms of that expression andµ µvar f ( ) = var[ f ′( p)( − p)]=nnµ p(1 − p)′ ′nn2 2[ f ( p)] var = [ f ( p)] .Let us choose the function f in such a manner that the expression forvar f(µ/n) would not depend on the unknown parameter p. Moreprecisely, we assume that′ − =2[ f ( p)] p(1 p) 1.This is a differential equation and we may choose any of its solutionsas f(p); in particular,f(p) = 2arcsin√p.Since, if allowing for the approximation made, f(µ/n) is a linearfunction of µ and, for large values of n, the distribution of µ isapproximately normal, the expressionµ µf ( ) = 2arcsin(4.8)nnis also approximately normal. Its expectation is approximatelyf(p) = 2arcsin√p37
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: along with ξ. For example, if ξ i
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
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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
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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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