Числові характеристики випадкової величини, пов'язаної з представленням дійсних чисел рядами Остроградського-Серпінського-Пірса
It is known that any irrational number $x\in\left(0;1\right)\backslash \mathbb{Q}\equiv\Omega$ has a unique Ostrogradsky-Sierpinski-Pierce expansion: $$x=\sum_{n=1}^{\infty}\frac{1}{q_1(x)\cdot...\cdot q_n(x)},$$ where $q_n(x)\in\mathbb{N}$, $q_{n+1}(x)> q_n(x)$, for all $n \in \mathbb{N}...
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| Datum: | 2019 |
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| Автори та афіліації: |
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| Ключові слова: | keywords |
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| Format: | Artikel |
| Sprache: | Ukrainisch |
| Veröffentlicht: |
Інститут математики НАН України
2019
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| Online Zugang: | https://trim.imath.kiev.ua/index.php/trim/article/view/514 |
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| Назва журналу: | Transactions of Institute of Mathematics of NAS of Ukraine |
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Transactions of Institute of Mathematics of NAS of Ukraine| Zusammenfassung: | It is known that any irrational number $x\in\left(0;1\right)\backslash \mathbb{Q}\equiv\Omega$ has a unique Ostrogradsky-Sierpinski-Pierce expansion: $$x=\sum_{n=1}^{\infty}\frac{1}{q_1(x)\cdot...\cdot q_n(x)},$$ where $q_n(x)\in\mathbb{N}$, $q_{n+1}(x)> q_n(x)$, for all $n \in \mathbb{N}$.To represent an irrational number $x\in\Omega$ by Ostrogradsky-Serpinsky-Pierce expansion we have calculated numerical characteristics of the random variable $$\xi(X)=\sum_{n=1}^{\infty}\frac{1}{q_n(X)},$$ where $X$ is uniform distribution on $\Omega$. A new method for calculating the mathematical expectation is proposed, which differs from the method described in \cite{Shallit1986}, and we have calculated variance $D\xi$. We consider the random variables $\xi_n$ as ageneralization of the function $\xi$ and we have calculated mathematical expectations $M\xi_n$ of them. |
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