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An Alternative Proof of Gauss’s Inequalities

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Cybernetics and Systems Analysis Aims and scope

A clear formulation of two Gauss’s inequalities is given, and their transparent proof based on the well-known fundamental results is presented. A simple method of constructing a partition of the parameter domain of the problem is proposed. An explicit form of the extreme distribution functions is found.

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References

  1. S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience Publ. (1966).

  2. R. E. Barlow and F. Proschan, Mathematical Theory of Reliability, John Wiley and Sons (1965).

  3. N. L. Johnson and C. A. Rogers, “The moment problem for unimodal distribution,” Ann. Math. Stat., Vol. 22, 433–439 (1951).

    Article  MathSciNet  MATH  Google Scholar 

  4. H. P. Mulholland and C. A. Rogers, “Representation theorems for distribution functions,” Proc. London Math. Soc., Vol. 3, No. 8, 177–223 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. G. Krein and A. A. Nudel’man, The Markov Moment Problem and Extremal Problems, in: Translations of Mathematical Monographs, Vol. 50 (1977).

  6. L. S. Stoikova and L. V. Kovalchuk, “Exact estimates for some linear functionals of unimodal distribution functions under incomplete information,” Cybern. Syst. Analysis, Vol. 55, No. 6, 914–925 (2019). https://doi.org.10.1007.s10559-019-00201-z.

  7. C. F. Gauss, “Theoria combination observation,” Werk, Goettingen, No. 4, 10–11 (1880).

    Google Scholar 

  8. B. H. Camp, “A new generalization of Thebyscheff’s statistical inequality,” Bull. Amer. Math. Soc., Vol. 28, 427–432 (1922).

    Article  MathSciNet  MATH  Google Scholar 

  9. B. Meidell, “Sur une problème du calcul des probabilités et les statistiques mathématiques,” C. R. Acad. Sci., Vol. 175, 806–808 (1922).

    MATH  Google Scholar 

  10. M. Fréchet, Généralités sur les Probabilités. Eléments Aléatoires, Borel Senes, in: Traité du Calcul des Probabilitéset de ses Applications, Div. 1, Pt. III, Vol. 1, Gauthier–Villars, Paris (1950).

  11. L. S. Stoikova, “Exact estimates of the probability of a non-negative unimodal random value hitting special intervals under incomplete information,” Cybern. Syst. Analysis, Vol. 57, No. 2, 264–267 (2021). https://doi.org.10.1007.s10559-021-00351-z.

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Correspondence to L. S. Stoikova.

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Translated from Kibernetyka ta Systemnyi Analiz, No. 2, March–April, 2023, pp. 64–71.

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Stoikova, L.S. An Alternative Proof of Gauss’s Inequalities. Cybern Syst Anal 59, 231–237 (2023). https://doi.org/10.1007/s10559-023-00557-3

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