kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

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Axiom 2.3 of realization. When proposition A of a given totality H is realized, anyproposition , having previously been a particular case of A, acquires in the transformedtotality a probability that depends only on the probabilities of A and in the given totality H.Thus, by definition, the probability A of the proposition after A was realized is A = f(Prob ; Prob A). (18)The definition of the realization of proposition A, as provided in §1.3.3, determined only thelogical structure of the transformed totality; and, since this does not completely determine theprobabilities of propositions, our new assumption 13 cannot be a corollary of the previousones.Let us now show that, if only the function f in formula (18) is 14 A = Prob /Prob A, (19)the axiom of realization does not contradict those assumed previously. Indeed, since thestructure of the propositions remaining in the transformed totality is the same as it was in theinitial one,f(Prob + Prob ; Prob A) = f(Prob ; Prob A) + f(Prob ; Prob A ).However, as is well known 15 , for this relation to hold it is necessary thatf(Prob ; Prob A) = (Prob ) F(Prob A).On the other hand, A A = 1 in accord with the condition; therefore,(Prob A)F(Prob A) = 1and (19) holds. At the same time we see that the function A determined by us implies that allthe propositions of the transformed totality acquire probabilities satisfying the condition ofarithmetization (§2.1.6) if only the probabilities of the propositions in the initial totality obeythe same condition. We see therefore that this function does not contradict the main axioms.2.2.4. The Multiplication TheoremTheorem 2.4. The probability of (A and B) is equal to the probability of A multiplied bythe probability of B given that A has occurred.Indeed, after A is realized, the proposition (A and B) is equivalent to( and B) = B andProb (A and B) = (A and B) A = B A = Prob (A and B)/Prob A.Consequently, Prob (A and B) = (Prob A) B A , QED.Note. The multiplication theorem implies, specifically, that if is a particular case of A,its probability only depends on the probability of A and on the probability of after A hasoccurred. Or, in an equivalent form: If and - are particular cases of A and B respectively,then their probabilities are equal to each other if A and B are equally probable and theprobability of after A has occurred is equal to the probability of after B has occurred.This statement can replace the axiom of realization because the multiplication theorem canbe derived therefrom similarly to the above. The introduction of the notion of probability of a
proposition given that a second one has occurred allows us to offer another definition ofindependence.If the probability of B after A has occurred is equal to its initial probability, then B isindependent of A.Corollary 2.6. If B is independent of A, then A is independent of B andProb (A and B) = Prob AProb B.2.2.5. The Bayes TheoremTheorem 2.5. The probability of A after B has occurred isA B = (Prob A)B A /Prob B.Indeed,A B = Prob (A and B)/Prob B = (Prob A) B A /Prob B.The axiom of realization (§2.2.3) is the sole basis of the Bayes theorem and its corollarieswhose derivation presents no difficulties in principle.Chapter 3. Infinite Totalities of Propositions3.1. Extension of the Preliminary Axioms onto Infinite Totalities3.1.1. Perfect TotalitiesThe main condition that we ought to introduce here is that the rules of the symboliccalculus established for finite totalities (§1.1) must persist should we consider thepropositions of a finite totality as belonging to some infinite totality. A totality (finite orotherwise) of propositions to which all these rules are applicable is called perfect. However,some of the premises, which were previously corollaries of the other ones, become here, forinfinite totalities, new independent axioms.Indeed, the existence of a true proposition, which was the corollary of axioms (a) – (d) of§1.1.2, is such a new assumption. Let us consider the totality of proper fractions p/2 n writtendown in the binary number system (0.101; 0.011; etc). We will understand the operation oras the compilation of a new fraction out of two given ones in such a manner that each of itsdigits equals the largest from among the two appropriate digits of the given fractions; thus,(0.101 or 0.011) = 0.111. This totality will not contain a number corresponding to the trueproposition that should have been represented by the infinite fraction 0.111… = 1. We can,however, add 1 to our fractions so as to realize the axiom of the true proposition. And, byjoining 0, we will also obtain the impossible proposition.The second necessary new assumption (a corollary of the previous premises for finitetotalities) is that the combination of two propositions (A and B) does exist. Finally, the thirdand last additional assumption is the extension of the restrictive principle onto infinite joinsof propositions.In the example concerning the binary fractions (with 0 and 1 being included) thecombination of two propositions existed and was represented by a fraction having as each ofits digits the least one from among the two appropriate digits of the given fractions. At thesame time, the restrictive principle is also applicable here to any pair of propositions so thatall the properties of combinations including the theorems of distributivity persist. In addition,the principle of uniqueness also holds in our example, but the totality considered will not beperfect. Indeed, in accord with the definition above (§1.1.8), a negation of any proposition Ais the join of all propositions incompatible with it. But there will be an infinite set of suchpropositions and we ought to begin by generalizing the definition of join as given in §1.1.1.
Page 3: of All Countries and to the Entire
Page 6 and 7: (Coll. Works), vol. 4. N.p., 1964,
Page 8 and 9: individuals of the third class, the
Page 10: From the theoretical point of view
Page 13 and 14: Second case: Each crossing can repr
Page 15 and 16: On the other hand, for four classes
Page 17 and 18: f i = i S + i , i = 1, 2, 3, 4, (
Page 19 and 20: f 1 = C 1 P(f 1 ; …; f n+1 ), C 1
Page 21 and 22: ut in this case f = 2 , f 1 = 2 ,
Page 23 and 24: I also note the essential differenc
Page 25 and 26: A 1 23n1 + 1 A 1 A 1 … A 11A 2 A
Page 27 and 28: coefficient of 2 in the right side
Page 29 and 30: h(A r h - c h A r 0 ) = - A r0we tr
Page 31 and 32: Notes1. Our formulas obviously pres
Page 33 and 34: Bernstein’s standpoint regarding
Page 35 and 36: Corollary 1.8. A true proposition c
Page 37 and 38: It is important to indicate that al
Page 39 and 40: ut for the simultaneous realization
Page 41 and 42: devoid of quadratic divisors and re
Page 43 and 44: propositions (B i and C j ) can be
Page 45 and 46: A ~ A 1 and B = B 1 , we will have
Page 47 and 48: included in a given totality as equ
Page 49: For unconnected totalities we would
Page 53 and 54: On the other hand, let x be a parti
Page 55 and 56: totality is perfect, but that the j
Page 57 and 58: In this case, all the finite or inf
Page 59 and 60: probabilities p 1 , p 2 , … respe
Page 61 and 62: where x is determined by the inequa
Page 63 and 64: totality of the second type (§3.1.
Page 65 and 66: x = /2 + /(23) + … + /(23… p n
Page 67 and 68: that the fall of a given die on any
Page 69 and 70: infinitely many digits only dependi
Page 71 and 72: 10. (§2.1.5). Such two proposition
Page 73 and 74: F(x + h) - F(x) = Mh, therefore F(x
Page 75 and 76: “confidence” probability is bas
Page 77 and 78: x1+ Lp n (x) x1− Lx1+ Lf(t)dt < x
Page 79 and 80: |(x 1 ; t 0 ; t 1 ) - 1 t0tf(t)dt|
Page 81 and 82: 5. The distribution ofξ , the arit
Page 83 and 84: P(x 1i < x) = F(x; a i ) = C(a i )
Page 85 and 86: egards his promises. Markov shows t
Page 87 and 88: other solely and equally possible i
Page 89 and 90: notion of probability and of its re
Page 91 and 92: However, already in the beginning o
Page 93 and 94: the revolution. My main findings we
Page 95 and 96: Nevertheless, Slutsky is not suffic
Page 97 and 98: path that would completely answer h
Page 99 and 100: on political economy as well as wit
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scientific merit. Borel was indeed
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[3] Already in Kiev Slutsky had bee
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different foundation. The difficult
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5. On the criterion of goodness of
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--- (1999, in Russian), Slutsky: co
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Here also, the author considers the
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second, it is not based on assumpti
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experimentation and connected with
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Russian, and especially of the Sovi
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station in England. This book, as h
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Uspekhi Matematich. Nauk, vol. 10,
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variety and detachment of those lat
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46. On the distribution of the regr
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119. On the Markov method of establ
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No lesser difficulties than those e
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Separate spheres of work considerab
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10. Anderson, O. Letters to Karl Pe
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Hier sind, im Allgemeinen, ganz ana
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Jedenfalls, glaube ich erwiesen zu
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werde ich das ganze Material in kur
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considered as the limiting case of
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and, inversely,] = m ...1 2 N[ ch h
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µ 2 2 = m 2 2 - 2m 2 m 1 2 + m 1 4
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(x k - x k+1 ) … (x k - x +) = E(
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the thus obtained relations as pert
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[1/S(S - 1)(S - 2)][(Si = 1Sx i ) 3
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( N −1)((S − N )(2NS− 3S− 3
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µ 5 + 2µ 2 µ 3 = U [S/S] 5 + 2U
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case, the same property is true wit
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It follows that the question about
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then expressed my doubts). And Gned
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For Problem 1, formula (7) shows th
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Let us calculate now, by means of f
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ϕ′1(x)1E(a|x 1 ; x 2 ; …; x n
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Theorem 3. If the prior density 3
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P( ≤ ≤ |, 1 , 2 , …, s )
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6. A Sensible Choice of Confidence
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0 = A 0 n, = B2, = B2, 0 = C 0 n
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Note also that (95),(96), (83),(85)
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Γ(n / 2)Γ [( n −1) / 2]k = (1/2
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f (x 1 , x 2 , …, x n ) = 1 if x
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and the probability of achieving no
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E = kEµ. (14)In many particular ca
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a = np, b = np 2 = a 2 /n, = a/nand
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with number (2k - 2), we commit an
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(67)which is suitable even without
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" = 1/[1 - e - ], = - ln [1 - (1/
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Such structures are entirely approp
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11. As a result of its historical d
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exaggeration towards a total denial
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kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

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