| Project/Area Number |
14540114
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kansai University (2004-2005)
Osaka University (2002-2003) |
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Principal Investigator |
AKI Sigeo Kansai University, Faculty of Engineering, Professor, 工学部, 教授 (90132696)
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| Co-Investigator(Kenkyū-buntansha) |
谷口 正信 大阪大学, 大学院・基礎工学研究科, 助教授 (00116625)
稲垣 宣生 大阪大学, 大学院・基礎工学研究科, 教授 (10000184)
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| Project Period (FY) |
2002 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2005: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Keywords | discrete patterns / discrete distribution theory / probability generating function / Markov chain / urn models / system reliability / waiting time problem / conditional expectation / 条件付き確率 / 離散確率分布 / 起動試験 / 工学的システムの信頼性 / start-up demonstration test / 幾何分布 / 条件付確率 / 壷のモデル |
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| Research Abstract |
We have studied the distributions of numbers of occurrences of discrete patterns and waiting time distributions for them in various dependent sequences. The following results have been derived.
1. By using a combinatorial method it is shown that for every finite pattern, the distribution of the waiting time for the reversed pattern coincides with that of the waiting time for the original pattern in a multi-state dependent sequence with a certain type of exchangeability. The corresponding results for the waiting time for the r-th occurrence of the pattern, and for the number of occurrences of a specified pattern in n trials are also studied. Illustrative examples based on urn models are also given.
2. Several waiting time random variables for a duplication within a memory window of size k in a sequence of {1,2,...,m}-valued random variables are investigated. The exact distributions of the waiting time random variables are derived by the method of conditional probability generating functio
… More
ns. In particular, the exact distribution of the waiting time for the first k-match is obtained when the underlying sequence is generated by higher order Markov dependent trials. Examples for numerical calculations are also given.
3. We consider waiting time problems for two-dimensional pattern in a sequence of i.i.d. random vectors each of whose entries is 0 or 1. 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.
4. Let k and m be positive integers with k>m. The probability generating function of the waiting time for the first occurrence of consecutive k successes in a sequence of m-th order Markov dependent trials is given as a function of the conditional probability generating functions of the waiting time for the first occurrence of consecutive m successes. This provides an efficient algorithm for obtaining the probability generating function when k is large. In particular, in the case of independent trials a simple relationship between the geometric distribution of order k and the geometric distribution of order k-1 is obtained.
5. Various new distributions related to runs and patterns in dependent sequences are obtained by using the stepwise smoothing formula for conditional expectations. Less
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