kniga 7 - Probability and Statistics 1 - Sheynin, Oscar

U 4 ]N[ + 3U 22 ]N[ = [(N – 1)/N] (µ 4 + 3µ 2 2 )or, when passing on to a uniform system, ofU 4 [N] + 3U 22 [N] = [(N – 1)/N] (M 4 + 3M 22 ).Supposing that N = S, we indeed find that(µ 4 + 3µ 2 2 ) = U 4 [S/S] + 3U 22 [S/S] = [(S – 1)/S] (M 4 + 3M 22 ),S [–2] (M 4 + 3M 22 ) = S 2 (µ 4 + 3µ 2 2 ).12. The pattern of an unreturned ticket is contrary to that of an attached ticket just as, inthe Lexian terminology, a scheme of a supernormally stable statistical series is opposed tothe arrangement of a series with a subnormal stability.13. The uniformity of the system of trials obtained on the basis of the new layout is verysimply revealed by means of a test established in the next section.14. Magnitudes x (i1) , x (i2) , …, x (iN) represent some of the values x (1) , x (2) , …, x (k) ; some or allof them may be identical. It is assumed here as it was before that the law of distribution ofthe variable remains fixed.References1. Bortkiewicz, L., von (1917), Die Iterationen. Berlin.2. Chuprov, A.A. (1918 – 1919), Zur Theorie der Stabilität statistischer Reihen. Skand.Aktuarietidskr., t. 1, pp. 199 – 256; t. 2, pp. 80 – 133.3. --- (1918 – 1919 and 1921), On the mathematical expectation of the moments offrequency distributions. Biometrika, vol. 12, pp. 140 – 169 and 185 – 210; vol. 13, pp. 283 –295.4. --- (1922), Ist die normale Dispersion empirisch nachweisbar? Nordisk Stat. Tidskr., t. 1,pp. 369 – 393.5. --- (1923), On the mathematical expectation of the moments of frequency distributionsin the case of correlated observations. Metron, t. 2, pp. 461 – 493 and 646 – 683.6. Seneta, E. (1987), Chuprov on finite exchangeability, expectation of ratios and measuresof association. Hist. Math., vol. 14, pp. 243 – 257.7. Sheynin, O. (1990, in Russian), Chuprov. Göttingen, 1996.12. A.N. Kolmogorov. Determining the Center of Scatteringand the Measure of Precision Given a Restricted Number of ObservationsIzvestia Akademii Nauk SSSR, ser. Math., vol. 6, 1942, pp. 3 – 32Foreword by TranslatorThis paper was apparently written hastily, and, at the time, its subject-matter did notperhaps belong to the author’s main scientific field. He mixed up Bayesian ideas and theconcept of confidence intervals and in §4 he showed that the posterior distribution of aparameter was asymptotically normal without mentioning that this was due to the wellknownBernstein- von Mises theorem. Points of more general interest are Kolmogorov’sdebate with Bernstein on confidence probability and, in Note 11, a new axiom of the theory ofprobability. The author apparently set high store by artillery (even apart from ballistics) as afield of application for probability theory. Indeed, this is seen from Gnedenko’s relevantstatement [1, p. 211] which he inserted even without substantiating it, a fact about which I