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On a general probability theorem and its applications in the theory of the stochastic processes

Published online by Cambridge University Press:  24 October 2008

Lajos Takács  and
W. L. Smith
Lajos Takács
Affiliation:
Research Institute for Mathematics, Hungarian Academy of Sciences, Budapest, Hungary
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Extract

Let us consider an experiment, the possible outcomes of which are random events. Let A1, A2, …, An be some of these outcomes. Define the random variable ηn as the number of the events occurring simultaneously among the A1, A2, …, An in an experiment. Denote by Pk = (ηn = k) (k = 0,1, 2, …, n), the distribution of the random variable ηn and by , the binomial moments of ηn. Here is the symbol of the probability and & that of the expectation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 54 , Issue 2 , April 1958 , pp. 219 - 224
Copyright
Copyright © Cambridge Philosophical Society 1958

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References

REFERENCES

(1)Darling, D. A.On a class of problems related to the random division of an interval. Ann. Math. Statist. 24 (1953), 239–53.Google Scholar
(2)Domb, C.The problem of random intervals on a line. Proc. Camb. Phil. Soc. 43 (1947), 329–41.Google Scholar
(3)Fisher, R. A.On the similarity of the distributions found for the test of significance in harmonic analysis and in Stevens's problem in geometric probability. Ann. Eugen., Lond., 10 (1940), 14–17.CrossRefGoogle Scholar
(4)Fortet, R.Calcul des probabilités (Paris, 1950).Google Scholar
(5)Garwood, E.An application of the theory of probability to the operation of vehicular-controlled traffic signals. Suppl. J. R. Statist. Soc. 7 (1940), 65–77.CrossRefGoogle Scholar
(6)Hammersley, J. M.On counters with random dead time. I. Proc. Camb. Phil. Soc. 50 (1954), 623–37.Google Scholar
(7)Jordan, C.A valószinüsegszámitás alapfogalmai. (Les fondements du calcul des prob-abilités.) Math. phys. Lapok, 34 (1927), 109–36.Google Scholar
(8)Jordan, C.Problèmes de la probabilité des éprewues répétées dans le cas général (Paris, 1939).Google Scholar
(9)Jordan, C.Calculus of finite differences (Budapest, 1939).Google Scholar
(10)Levert, C. and Scheen, W. L.Probability fluctuation of discharges in a Geiger-Müller counter produced by cosmic radiation. Physica, 10 (1943), 225–38.Google Scholar
(11)Lévy, P.Sur la division d'un segment par des points choisis au hasard. C.R. Acad. Sci., Paris, 208 (1939), 147–49.Google Scholar
(12)Morgan, P.The random division of an interval. Suppl. J. R. Statist. Soc. 9 (1947), 92–8.Google Scholar
(13)Ramakrishnan, A.Stochastic processes associated with random divisions of a line. Proc. Camb. Phil Soc. 49 (1953), 473–85.Google Scholar
(14)Stevens, W. L.Solution to a geometrical problem in probability. Ann. Eugen., Lond., 9 (1939), 315–20.Google Scholar
(15)Takács, L.On processes of happenings generated by means of a Poisson-process. Acta Math. Acad. Sci. Hung. 6 (1955), 81–99.CrossRefGoogle Scholar
(16)Takács, L.On a probability problem arising in the theory of counters. Proc. Camb. Phil. Soc. 52 (1956), 488–498.Google Scholar
(17)Whitworth, W. A.Choice and chance (Cambridge, 1897).Google Scholar
(18)Fréchet, M.Les probabilités associées á un système d'événements compatibles et dépendants (Paris, 1940).Google Scholar
(19)Chung, K. L.On the probability of the occurrence of at least m events among n arbitrary events. Ann. Math. Statist. 12 (1941), 328–38.Google Scholar