The average position of the first maximum in a sample of geometric random variables
We consider samples of n geometric random variables $(Γ _1, Γ _2, \dots Γ _n)$ where $\mathbb{P}\{Γ _j=i\}=pq^{i-1}$, for $1≤j ≤n$, with $p+q=1$. The parameter we study is the position of the first occurrence of the maximum value in a such a sample. We derive a probability generating function for th...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2007-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/3523/pdf |
| _version_ | 1827324046813954048 |
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| author | Margaret Archibald Arnold Knopfmacher |
| author_facet | Margaret Archibald Arnold Knopfmacher |
| author_sort | Margaret Archibald |
| collection | DOAJ |
| description | We consider samples of n geometric random variables $(Γ _1, Γ _2, \dots Γ _n)$ where $\mathbb{P}\{Γ _j=i\}=pq^{i-1}$, for $1≤j ≤n$, with $p+q=1$. The parameter we study is the position of the first occurrence of the maximum value in a such a sample. We derive a probability generating function for this position with which we compute the first two (factorial) moments. The asymptotic technique known as Rice's method then yields the main terms as well as the Fourier expansions of the fluctuating functions arising in the expected value and the variance. |
| first_indexed | 2024-04-25T02:03:38Z |
| format | Article |
| id | doaj.art-1743f23daa3444d5b5a2db0faf99a1a0 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:03:38Z |
| publishDate | 2007-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-1743f23daa3444d5b5a2db0faf99a1a02024-03-07T14:34:52ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502007-01-01DMTCS Proceedings vol. AH,...Proceedings10.46298/dmtcs.35233523The average position of the first maximum in a sample of geometric random variablesMargaret Archibald0Arnold Knopfmacher1Department of Mathematics and Applied Mathematics [Cape Town]The John Knopfmacher Centre for Applicable Analysis and Number Theory [Johannesburg]We consider samples of n geometric random variables $(Γ _1, Γ _2, \dots Γ _n)$ where $\mathbb{P}\{Γ _j=i\}=pq^{i-1}$, for $1≤j ≤n$, with $p+q=1$. The parameter we study is the position of the first occurrence of the maximum value in a such a sample. We derive a probability generating function for this position with which we compute the first two (factorial) moments. The asymptotic technique known as Rice's method then yields the main terms as well as the Fourier expansions of the fluctuating functions arising in the expected value and the variance.https://dmtcs.episciences.org/3523/pdfgeometric random variablerice's method[info.info-ds] computer science [cs]/data structures and algorithms [cs.ds][info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co][info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| spellingShingle | Margaret Archibald Arnold Knopfmacher The average position of the first maximum in a sample of geometric random variables Discrete Mathematics & Theoretical Computer Science geometric random variable rice's method [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| title | The average position of the first maximum in a sample of geometric random variables |
| title_full | The average position of the first maximum in a sample of geometric random variables |
| title_fullStr | The average position of the first maximum in a sample of geometric random variables |
| title_full_unstemmed | The average position of the first maximum in a sample of geometric random variables |
| title_short | The average position of the first maximum in a sample of geometric random variables |
| title_sort | average position of the first maximum in a sample of geometric random variables |
| topic | geometric random variable rice's method [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| url | https://dmtcs.episciences.org/3523/pdf |
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