Abstract

Extensions of S.N. Bernstein's example of three dependent random events of which any two are independent are considered. A complete description of such examples is given in the framework of the symmetric probability model.

Keywords

  1. Bernstein example
  2. dependent (independent) random events
  3. Diophantine equation
  4. $j$-free number

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References

1.
S. Bernstein, Probability Theory, Gostekhizdat, Moscow, 1946 (in Russian).
2.
A. A. Borovkov, Probability Theory, Gordon and Breach Sci. Publ., Amsterdam, 1998.
3.
B. W. Gnedenko, Einführung in die Wahrscheinlichkeitstheorie, Math. Lehrbücher Monogr. I. Abt. Math. Lehrbücher 39, Akademie-Verlag, Berlin, 1991.
4.
G. J. Székély, Paradoxes in Probability Theory and Mathematical Statistics, Math. Appl. (East Eur. Ser.) 15, D. Reidel Publishing, Dordrecht, 1986.
5.
J. M. Stoyanov, Counterexamples in Probability, 2nd ed., Wiley Ser. Probab. Stat., John Wiley & Sons, Chichester, 1997.
6.
A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika, 43 (1996), pp. 73--107, https://doi.org/10.1112/S0025579300011608.
7.
A. A. Bukhshtab, Number Theory, Prosveshchenie, Moscow, 1966 (in Russian).

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Published In

cover image Theory of Probability & Its Applications
Theory of Probability & Its Applications
Pages: 318 - 325
ISSN (online): 1095-7219

History

Submitted: 12 September 2019
Published online: 5 August 2021

Keywords

  1. Bernstein example
  2. dependent (independent) random events
  3. Diophantine equation
  4. $j$-free number

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