On generalizations of the optimal choice problem
DOI:
https://doi.org/10.21638/spbu01.2021.103Abstract
The article is dedicated to some generalizations of the classical optimal choice problem (the fastidious bride problem, the secretary problem). Let there be a sequence of n identically distributed random variables on the interval [0, 1]. Getting consistently observed values of these variables, we should stop at some moment on one of them, accepting it as the initial point for counting upper or lower record values. In the optimal choice problem and its generalizations, it is needed to make the correct choice of the initial point of counting records, in order to guess the place of the last record (the classical optimal choice problem) or to maximize the expected sum of upper and/or lower record values or the expected total number of upper and/or lower records, obtained by this procedure. A review of results on the uniform distribution of the random variables and some new results concerning the exponential distribution are presented.Keywords:
record moments, record values, sums of record values, mean number of records, uniform distribution, exponential distribution, optimal choice problem
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References
Литература
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6. Бельков И.В. О некоторых задачах оптимального выбора рекордных величин. Научный журнал Globus, вып. 11 (44), 46–49 (2019).
References
1. Gardner M. Mathematical Games. A fifth collection of “brain-teasers”. Scientific American 202 (2), 150–154 (1960).
2. Dynkin E.B. Optimal choice of the stopping time of a Markov process. Dokl. Akad. Nauk SSSR 150 (2), 238–240 (1963). (In Russian) [Engl. transl.: Soviet Math. Dokl. 4, 627–629 (1963)].
3. Nevzorov V.B., Tovmasyan S.A. On the maximal value of the expectation of record numbers. Vestnik of Saint Petersburg University. Series 1. Mathematics. Mechanics. Astronomy 1 (59), iss. 2, 196–200 (2014). (In Russian) [Engl. transl.: Vestnik St. Petersb. Univ. Math. 47, iss. 2, 64–67 (2014). https://doi.org/10.3103/S1063454114020046].
4. Belkov I.V., Nevzorov V.B. On the problem of the optimal choice of record values. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 5 (63), iss. 2, 179–188 (2018). https://doi.org/10.21638/11701/spbu01.2018.201 (In Russian) [Engl. transl.: Vestnik St. Petersb. Univ. Math. 51, iss. 2, 107–113 (2018). https://doi.org/10.3103/S1063454118020024].
5. Belkov I.V., Nevzorov V.B. On one problem of the optimal choice of record values. Zap. Nauchn. Sem. POMI 466, 30–37 (2017). (In Russian) [Engl. transl.: J. Math. Sci. 244, 718–722 (2020). https://doi.org/10.1007/s10958-020-04644-0].
6. Belkov I.V. About some tasks of optimal selection of record values. Scientific journal “Globus”, iss. 11 (44), 46–49 (2019). (In Russian)
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Published
2021-05-29
How to Cite
Belkov, I. V. (2021). On generalizations of the optimal choice problem. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(1), 29–36. https://doi.org/10.21638/spbu01.2021.103
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
