kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

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This is the formula that attaches especial simplicity to the expression Eµ. Unlike thismagnitude, the probabilities P m and R m cannot be uniquely represented in the general case bythe probabilities p i = P(B i ). For determining these, it is necessary, generally speaking, toknow, in addition to the probabilities p i of each B i , the essence of the dependence betweenthe events B i . Here, we consider the case of mutually independent random events B 1 , B 2 , …,B n so that the probability that hits occur as a result of shots i 1 and do not occurin any other shot is 5pi1 i2im∏iqi... q1 2 i j mqp... pq j .Adding all such products corresponding to a given m, we will indeed determine theprobabilityP m = ( …pi1 i2im)∏iqi... q1 2 i jmqp... pq j , i 1 < i 2 < … < i m (22)of achieving exactly m hits after n shots. In particular,P 0 =∏jq j , P 1 = (ipi ) ∏qijq j , P 2 = ( pqi1i1pqi2i2)∏jq j , i 1 < i 2 .The number of terms in brackets in (22) is equal tomCn. Therefore, ifp 1 = p 2 = … = p n = p, (23)mwe arrive at the well-known formula P m = Cnp m q n–m , q = 1 – p.The general formula (22) is too complicated for application without extreme necessity.When the probabilities p i of hit for each separate shot are sufficiently low, it can be replacedby much more convenient appropriate formulas. These are the more exact, the lower is theupper bound = max (p 1 ; p 2 ;…; p n ) of the probabilities p i . The simplest 6 among them is thePoisson formulaP m = (a m /m!)e –a + O( ), a = p 1 + p 2 + + … + p n = Eµ. (27)This formula represents the probability P m to within the remainder term O( ). Thepublished tables 8 of the function (m; a) = (a m /m!)e –a make its practical application simpler.A more complicated but at the same time a more precise formula 7 can be recommended:P m = (m; a) – (b/2) ∇ 2 (m; a) + O( 2 ) (30)whereb = p 1 2 + p 2 2 + + … + p n 2 , (31)∇ 2 (m; a) = (m; a) – 2(m – 1; a) + (m – 2; a). 9The remainder term in (30) has order 2 .When all the probabilities are identical (23),
a = np, b = np 2 = a 2 /n, = a/nand formulas (27) and (30) are reduced to 10P m = (m; a) + O(1/n), P m = (m; a) – (a 2 /2n) ∇ 2 (m; a) + O(1/n 2 ). (33; 34)To show that the refined formulas (31) and (34) are advantageous as compared with (27)and (33), let us consider the following example: n = 5, p 1 = p 2 = … = p 5 = 0.3, a = 1.5 and b= 0.45. The corresponding values of P) m = (m; a), P+ m = (m; a) – (b/2) ∇ 2 (m; a) and P mare given in Table 1 {omitted}.Since the probabilities p i are here still rather large, the Poisson formula (27) or the formula(33) only offer a very rough approximation P) m to the true values of P m but the refinedformulas (30) and (34) already provide the approximations P+ m which differ from P m lessthan by 0.015.Let us also consider the case n = 50, p 1 = p i = 0.03, i = 1, 2, …, 50, a = 1.5, b = 0.045. Thecorresponding values of P m , P) m and P+ m are adduced in Table 2 {omitted} to within 0.00001.Here, even the usual Poisson formula provides admissible results (the deviations of P) m fromP m are less than 0.005) whereas the refined formulas (30) and (34) furnish P m with an errorless than 0.0001.For the probabilities R m of achieving not less than m hits, formulas (27), (30), (33) and(34), when issuing from (3), lead toR m = H(m; a) + O( ), R m = H(m; a) – (b/2) ∇ 2 H(m; a) + O( 2 ), (35; 36)R m = H(m; a) + O(1/n), R m = H(m; a) – (a 2 /2n) ∇ 2 H(m; a) + O(1/n 2 ) (38a; b)where, assuming that H(m; a) = 1 for m < 1,H(m; a) = 1 – (0; a) – (1; a) – … – (m – 1; a).The tables ofH(1; a) = 1 – e –a and H(2; a) = 1 – (1 + a)e –aare available in a number of treatises on the theory of firing 11 . The values of H(m; a) and∇ 2 H(m; a) = – ∇ (m – 1; a) (41)for m ≤ 11 are adduced in Tables 1 and 2 of the Supplement to this book {to the originalRussian source} and also there see Fig. 1. By means of these tables the probabilities R m aredetermined in accord with formulas (36) and (38) with a very small work input. Suppose forexample that n = 24 andp 1 = p 2 = … = p 6 = 0.20, p 7 = p 8 = … = p 12 = 0.10, p 13 = p 14 = … = p 18 =0.15 and p 19 = p 20 = … = p 24 = 0.05.It is required to determine the probability R 3 of achieving not less than three hits. We have a= p i = 3.00, b = p i 2 = 0.45. According to the tables, for m = 3 and a = 3, H = 0.577 and∇ 2 H = 0.075. Substituting these values in formula (36) results inR 3 0.577 + (0.45/2)0.075 = 0.594.
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of All Countries and to the Entire
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(Coll. Works), vol. 4. N.p., 1964,
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individuals of the third class, the
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From the theoretical point of view
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Second case: Each crossing can repr
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On the other hand, for four classes
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f i = i S + i , i = 1, 2, 3, 4, (
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f 1 = C 1 P(f 1 ; …; f n+1 ), C 1
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ut in this case f = 2 , f 1 = 2 ,
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I also note the essential differenc
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A 1 23n1 + 1 A 1 A 1 … A 11A 2 A
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coefficient of 2 in the right side
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h(A r h - c h A r 0 ) = - A r0we tr
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Notes1. Our formulas obviously pres
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Bernstein’s standpoint regarding
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Corollary 1.8. A true proposition c
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It is important to indicate that al
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ut for the simultaneous realization
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devoid of quadratic divisors and re
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propositions (B i and C j ) can be
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A ~ A 1 and B = B 1 , we will have
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included in a given totality as equ
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For unconnected totalities we would
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proposition given that a second one
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On the other hand, let x be a parti
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totality is perfect, but that the j
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In this case, all the finite or inf
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probabilities p 1 , p 2 , … respe
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where x is determined by the inequa
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totality of the second type (§3.1.
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x = /2 + /(23) + … + /(23… p n
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that the fall of a given die on any
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infinitely many digits only dependi
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10. (§2.1.5). Such two proposition
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F(x + h) - F(x) = Mh, therefore F(x
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“confidence” probability is bas
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x1+ Lp n (x) x1− Lx1+ Lf(t)dt < x
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|(x 1 ; t 0 ; t 1 ) - 1 t0tf(t)dt|
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5. The distribution ofξ , the arit
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P(x 1i < x) = F(x; a i ) = C(a i )
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egards his promises. Markov shows t
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other solely and equally possible i
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notion of probability and of its re
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However, already in the beginning o
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the revolution. My main findings we
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Nevertheless, Slutsky is not suffic
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path that would completely answer h
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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
Page 135 and 136: Hier sind, im Allgemeinen, ganz ana
Page 137 and 138: Jedenfalls, glaube ich erwiesen zu
Page 139 and 140: werde ich das ganze Material in kur
Page 141 and 142: considered as the limiting case of
Page 143 and 144: and, inversely,] = m ...1 2 N[ ch h
Page 145 and 146: µ 2 2 = m 2 2 - 2m 2 m 1 2 + m 1 4
Page 147 and 148: (x k - x k+1 ) … (x k - x +) = E(
Page 149 and 150: the thus obtained relations as pert
Page 151 and 152: [1/S(S - 1)(S - 2)][(Si = 1Sx i ) 3
Page 153 and 154: ( N −1)((S − N )(2NS− 3S− 3
Page 155 and 156: µ 5 + 2µ 2 µ 3 = U [S/S] 5 + 2U
Page 157 and 158: case, the same property is true wit
Page 159 and 160: It follows that the question about
Page 161 and 162: then expressed my doubts). And Gned
Page 163 and 164: For Problem 1, formula (7) shows th
Page 165 and 166: Let us calculate now, by means of f
Page 167 and 168: ϕ′1(x)1E(a|x 1 ; x 2 ; …; x n
Page 169 and 170: Theorem 3. If the prior density 3
Page 171 and 172: P( ≤ ≤ |, 1 , 2 , …, s )
Page 173 and 174: 6. A Sensible Choice of Confidence
Page 175 and 176: 0 = A 0 n, = B2, = B2, 0 = C 0 n
Page 177 and 178: Note also that (95),(96), (83),(85)
Page 179 and 180: Γ(n / 2)Γ [( n −1) / 2]k = (1/2
Page 181 and 182: f (x 1 , x 2 , …, x n ) = 1 if x
Page 183 and 184: and the probability of achieving no
Page 185: E = kEµ. (14)In many particular ca
Page 189 and 190: with number (2k - 2), we commit an
Page 191 and 192: (67)which is suitable even without
Page 193 and 194: " = 1/[1 - e - ], = - ln [1 - (1/
Page 195 and 196: Such structures are entirely approp
Page 197 and 198: 11. As a result of its historical d
Page 199 and 200: exaggeration towards a total denial
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kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

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