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Sooner and later waiting time problems based on a dependent sequence

ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 2000年第3期52卷 407-414页

作者： Han, Q Aki, S Osaka Univ Grad Sch Engn Sci Dept Informat & Math Sci Osaka 5608531 Japan

This paper introduces a new concept: a binary sequence of order (Tc, r), which is an extension of a binary sequence of order Ic and a Markov dependent sequence. The probability functions of the sooner and later waiting time random variables are derived in the binary sequence of order (Ic, r). The probability generating functions of the sooner and later waiting time distributions are also obtained. Extensions of these results to binary sequence of order (g, h) are also presented.

关键词： binary sequence of order (k, r) binary sequence of order k sooner and later waiting time problems probability function probability generating function

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DISTRIBUTIONS OF NUMBERS OF FAILURES AND SUCCESSES UNTIL THE 1ST CONSECUTIVE k SUCCESSES

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ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 1994年第1期46卷 193-202页

作者： AkI, S HIRANO, k INST STAT MATH MINATO KUTOKYO 106JAPAN

Exact distributions of the numbers of failures, successes and successes with indices no less than 1 (1 less-than-or-equal-to 1 less-than-or-equal-to k - 1) until the first consecutive k successes are obtained for some {0, 1}-valued random sequences such as a sequence of independent and identically distributed (iid) trials, a homogeneous Markov chain and a binary sequence of order k. The number of failures until the first consecutive k successes follows the geometric distribution with an appropriate parameter for each of the above three cases. When the {0, 1}-sequence is an iid sequence or a Markov chain, the distribution of the number of successes with indices no less than l is shown to be a shifted geometric distribution of order k - 1. When the {0, 1}-sequence is a binary sequence of order k, the corresponding number follows a shifted version of an extended geometric distribution of order k - 1.

关键词： GEOMETRIC DISTRIBUTION DISCRETE DISTRIBUTIONS MARkOV CHAIN WAITING TIME GEOMETRIC DISTRIBUTION OF order k IID sequence binary sequence of order k INVERSE SAMPLING

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Derivation of the probability distribution functions for succession quota random variables

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ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 1996年第3期48卷 551-561页

作者： Antzoulakos, DL Philippou, AN UNIV PATRAS DEPT MATHPATRAS 26110GREECE

The probability distribution functions (pdf's) of the sooner and later waiting time random variables (rv's) for the succession quota problem (k successes and r failures) are derived presently in the case of a binary sequence of order Ic. The probability generating functions (pgf's) of the above rv's are then obtained directly from their pdf's. In the case of independent Bernoulli trials, expressions for the pdf's in terms of binomial coefficients are also established.

关键词： succession quota sooner and later problems Bernoulli trials binary sequence of order k probability distribution function probability generating function longest run

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JOINT DISTRIBUTIONS OF NUMBERS OF SUCCESS-RUNS AND FAILURES UNTIL THE FIRST CONSECUTIVE k SUCCESSES

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ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 1995年第2期47卷 225-235页

作者： AkI, S HIRANO, k INST STAT MATH MINATO KUTOKYO 106JAPAN

Joint distributions of the numbers of failures, successes and success-runs of length less than k until the first consecutive k successes are obtained for some random sequences such as a sequence of independent and identically distributed integer valued random variables, a {0, 1}-valued Markov chain and a binary sequence of order k. There are some ways of counting numbers of runs with a specified length. This paper studies the joint distributions based on three ways of counting numbers of runs, i.e., the number of overlapping runs with a specified length, the number of non-overlapping runs with a specified length and the number of runs with a specified length or more. Marginal distributions of them can be derived immediately, and most of them are surprisingly simple.

关键词： PROBABILITY GENERATING FUNCTION GEOMETRIC DISTRIBUTION DISCRETE DISTRIBUTIONS MARkOV CHAIN WAITING TIME GEOMETRIC DISTRIBUTION OF order k binary sequence of order k

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Joint distributions of numbers of success-runs and failures until the first consecutive k successes in a binary sequence

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ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 1997年第3期49卷 519-529页

作者： Balakrishnan, N MCMASTER UNIV DEPT MATH & STATHAMILTONON L8S 4K1CANADA

Joint distributions of the numbers of failures, successes and success-runs of length less than k until the first consecutive k successes in a binary sequence were derived recently by Aki and Hirano (1995, Ann. Inst. Statist. Math., 47, 225-235). In this paper, we present an alternate derivation of these results and also use this approach to establish some additional results. Extensions of these results to binary sequences of order h are also presented.

关键词： probability generating function waiting time binary sequence of order k geometric distribution of order k

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