The number of distinct adjacent pairs in geometrically distributed words
A sequence of geometric random variables of length $n$ is a sequence of $n$ independent and identically distributed geometric random variables ($\Gamma_1, \Gamma_2, \dots, \Gamma_n$) where $\mathbb{P}(\Gamma_j=i)=pq^{i-1}$ for $1~\leq~j~\leq~n$ with $p+q=1.$ We study the number of distinct adjacent...
| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
2021-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/5686/pdf |
| Summary: | A sequence of geometric random variables of length $n$ is a sequence of $n$
independent and identically distributed geometric random variables ($\Gamma_1,
\Gamma_2, \dots, \Gamma_n$) where $\mathbb{P}(\Gamma_j=i)=pq^{i-1}$ for
$1~\leq~j~\leq~n$ with $p+q=1.$ We study the number of distinct adjacent two
letter patterns in such sequences. Initially we directly count the number of
distinct pairs in words of short length. Because of the rapid growth of the
number of word patterns we change our approach to this problem by obtaining an
expression for the expected number of distinct pairs in words of length $n$. We
also obtain the asymptotics for the expected number as $n \to \infty$. |
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| ISSN: | 1365-8050 |