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Комбинаторные методы в теории случайных процессов - Такач Л.

Такач Л. Комбинаторные методы в теории случайных процессов — М.: Мир, 1971. — 179 c.
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данных за А\, Аг Ап соответственно. Найти
Р [а^ ^ <4г) > ... > а*,г) для г = 1,2,..., а, + а2 + • • • + ап)
в предположении, что все избирательные протоколы равновероятны. (Макмагон
[26, стр. 133].)
ЛИТЕРАТУРА
[1] Aebly J., Demonstration du probleme du serutin par des considerations
geo-metriques, L'Enseignement Math., 23 (1923), 185-186.
[2] A e p p 1 i A., A propos de l'interpretation geometrique du probleme
du serutin, L'Enseignement Math., 23 (1923), 328-329.
[3] Aeppli A., Zur Theorie verketteter Wahrscheinlichkeiten, Markoffsche
Ketten hoherer Ordnung, Zurich, 1924.
[4] Ampere A. М., Considerations sur la theorie mathematique de Jeu,
Lyon, Paris, 1802.
[5] Andre D., Solution directe du probleme resolu par M. Bertrand, C. R.
Acad. Sci. Paris, 105 (1887), 436-437.
[6] В a r b i e г Ё., Generalisation du probleme resolu par M. J.
Bertrand, C. R. Acad. Sci. Paris, 105 (1887), 407, 440 (список опечаток).
[7] Barton D. E., M a 11 о w s C. L., Some aspects of the random
sequence, Ann. Math. Statist., 36 (1965), 236-260.
[8] Bertrand J., Solution d'un probleme, C. R. Acad. Sci. Paris, 105
(1887), 369.
[9] В i z 1 e у М. T. L., Derivation of a new formula for the number of
minimal lattice paths from (0,0) to (km,kn) having just t contacts with
the line my=nx and having no points above this line; and a proof of
Grossman's formula for the number of paths which may touch but do not
rise above this line, J. Inst. Actuar., 80 (1954), 55-62.
10] Bizley М. T. L., Grossman H. D" Fun with lattice-points 25. Paths
having a given number of lattice points in a given region, Scripta Math.,
20 (1954), 203-204.
11] D e Moivre A, De mensura sortis, seu, de probabilitate eventuum in
ludis a casu fortuito pendentibus, Philos. Trans. London, 27 (1711), 213-
264.
12] D i n g e s H., Eine kombinatorische tJberlegung und ihre
masstheoretische Er-weiterung, Z. Wahr., 1 (1963), 278-287.
13] DvoretzkyA., Motzkin Th., A problem of arrangements, Duke Math. J.,
14 (1947), 305-313.
14] Erdos P., Kaplansky I., Sequences of plus and minus, Scripta Math.,
12 (1946), 73-75.
15] Gob el F., Some remarks on ballot problems, Mathematisch Centrum,
Amsterdam, August, 1964.
16] Graham R. L., A combinatorial theorem for partial sums, Ann. Math.
Statist., 34 (1963), 1600-1602.
17] Grossman H. D., Fun with lattice-points 4, The ballot-box problem,
Scripta Math., 12 (1946), 223-225.
18] Grossman H. D., Fun with lattice-points 21, Another extension of the
ballot problem", Scripta Math., 16 (1950), 120-124.
19] Grossman H. D., Fun with lattice-points 22, Paths in a lattice
triangle, Scripta Math., 16 (1950), 207-212.
20] Harper L. H., A family of combinatorial identities, Ann. Math.
Statist., 37 (1966), 509-512.
21] Hewitt E., Stromberg K-, Real and abstract analysis, Springer, New
York, 1965.
22] Lagrange J. L., Recherches sur les suites recurrentes dont les termes
va-rient de plusieurs manieres, differentes, ou sur 1'integration des
equations
Литература
15
lineaires aux differences finies et partielles; et sur l'usage de ces
equations dans la theorie des hazards, Noveaux Memoires de l'Academie des
Sciences de Berlin, 1775, 183-272. (Oeuvres de Lagrange, IV, pp. 151-251,
Paris, 1869.)
[23] Laplace P. S., Theorie analytique des probabilites, Paris, 1812.
(Oeuvres completes de Laplace, VII, Paris, 1886.)
[24] Lucas E., Theorie des nombres. I, Paris, 1891.
[25] M а с M a h о n P. A., Memoir on the theory of the partitions of
numbers. Part IV. On the probability that the successful condidate at the
election by ballot may never at any time have fewer votes than the one
who is unsuccessful; on a generalisation of this question; and on its
connexion with other questions of partition, permutation and combination,
Phil. Trans. Roy. Soc. London, Ser. A, 209 (1909), 153-175.
[26] M а с M a h о n P. A., Combinatory analysis, I, Cambridge University
Press, 1915.
[27] Mirimanoff D., A propos de ('interpretation geometrique de probleme
du serutin, L'Enseignement Math., 23 (1923), 185-186.
[28] Mohanty S. Q., An urn problem related to the ballot problem, Amer.
Math. Monthly, 73 (1966), 526-528.
[29] Mohan t у S. G., N a г а у a n а Т. V., Some properties of
compositions and their application to probability and statistics,
Biometrische Z., 3 (1961), 252-258, 5 (1963), 8-18.
[30] N a г а у a n а Т. V., A partial order and its applications to
probability theory, Sankhya, Ser. A, 21 (1959), 91-98.
[31] R i о r d a n J., The enumeration of election returns by number of
lead positions, Ann. Math. Statist., 35 (1964), 369-379.
[32] S z. - N a g у В., Introduction to real functions and orthogonal
expansions, Oxford University Press, New York, 1965.
[33] T а к a с s L., A generalization of the ballot problem and its
application in the theory of queues, J. Amer. Statist. Assoc., 57 (1962),
327-337.
[34] Takacs L., Ballot problems, Z. Wahr., 1 (1962), 154-158.
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