| Project/Area Number |
16500183
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Statistical science
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| Research Institution | The Institute of Statistical Mathematics |
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Principal Investigator |
HIRANO Katuomi The Institute of Statistical Mathematics, Department of Mathematical Analysis and Statistical Inference, Professor (30000186)
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| Project Period (FY) |
2004 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,240,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2005: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | Waiting time problems / Probability generating function / Markov chain / Conditional expectation / Pattern / Reliability / Consecutive system / Geometric distribution of order k / パターン / 条件付期待値 / exchangeable sequence / 統計数学 / 確率論 / 文字列 / 連 / 離散分布 / システムの信頼性 |
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| Research Abstract |
[1]. We consider waiting time problems for a two-dimensional pattern in a sequence of i.i.d. random vectors each of whose entries is 0 or 1. We deal with a two-dimensional pattern with a general shape in the two-dimensional lattice which is generated by the above sequence of random vectors. A general method for obtaining the exact distribution of the waiting time for the first occurrence of the pattern in the sequence is presented. The method is an extension of the method of conditional probability generating functions and it is very suitable for computations with computer algebra systems as well as usual numerical computations. Computational results applied to computation of exact system reliability are also given.
[2]. A simple relationship between the geometric distributions of order (k-1) and of order k is obtained based on stepwise conditioning with an increasing sequence of stepping times. The idea is extended to the corresponding problem for higher order Markov dependent trials a
… More
nd the probability generating function of the waiting time for the first occurrence of consecutive k successes in the m-th order Markov dependent trials is given explicitly. This provides surprisingly good algorithm for obtaining the probability generating function when k is large.
[3]. This paper proposes a method for obtaining the exact probability of occurrence of the first success run of specified length with the additional constraint that at every trial until the occurrence of the first success run the number of successes up to the trial exceeds that of failures. For the sake of the additional constraint, the problem can not be solved by the usual method of conditional probability generating functions. An idea of a kind of truncation is introduced and studied in order to solve the problem. Concrete methods for obtaining the probability in the cases of Bernoulli trials and time-homogeneous{0, 1}-valued Markov dependent trials are given. As an application of the results, a modification of the start-up demonstration test is studied. Numerical examples which illustrate the feasibility of the results are also given.
[4]. Exact joint distributions of waiting times for two patterns in a sequence of higher order time-homogeneous Markov dependent trials are studied, where the patterns are not necessarily assumed to be distinct with each other. We prove that exact joint probability generating functions, which are regarded as expectations of the corresponding random variables, are derived through calculating the conditional expectation based on conditioning by the sooner waiting time and the pattern which comes sooner. We also give illustrative numerical examples in order to demonstrate the performance of our results. Less
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