Rearrangements of distributions on integers that minimize variance
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909992924217344 |
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| author | Atminas, Aistis Kurauskas, Valentas |
| author_facet | Atminas, Aistis Kurauskas, Valentas |
| contents | Which permutations of a probability distribution on integers minimize variance?
Let $X$ be a random variable on a set of integers $\{x_1, \dots, x_N\}$ such that $\mathbb{P}(X_i = x_i) = p_i$, $i \in \{1,\dots,N\}$. Let $(p^{(1)}, \dots, p^{(N)})$ be the sequence $(p_1, \dots, p_N)$ ordered non-increasingly. Let $X^+$ be the random variable defined by $\mathbb{P}(X^+=0)=p^{(1)}$, $\mathbb{P}(X^+=1) = p^{(2)}$, $\mathbb{P}(X^+=-1)=p^{(3)}, \dots, \mathbb{P}(X^+=(-1)^N \lfloor \frac {N} 2 \rfloor)=p^{(N)}$. In this short note we generalize and prove the inequality $\mathrm{Var}\, X^+ \le \mathrm{Var}\, X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07899 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rearrangements of distributions on integers that minimize variance Atminas, Aistis Kurauskas, Valentas Combinatorics Probability Which permutations of a probability distribution on integers minimize variance? Let $X$ be a random variable on a set of integers $\{x_1, \dots, x_N\}$ such that $\mathbb{P}(X_i = x_i) = p_i$, $i \in \{1,\dots,N\}$. Let $(p^{(1)}, \dots, p^{(N)})$ be the sequence $(p_1, \dots, p_N)$ ordered non-increasingly. Let $X^+$ be the random variable defined by $\mathbb{P}(X^+=0)=p^{(1)}$, $\mathbb{P}(X^+=1) = p^{(2)}$, $\mathbb{P}(X^+=-1)=p^{(3)}, \dots, \mathbb{P}(X^+=(-1)^N \lfloor \frac {N} 2 \rfloor)=p^{(N)}$. In this short note we generalize and prove the inequality $\mathrm{Var}\, X^+ \le \mathrm{Var}\, X$. |
| title | Rearrangements of distributions on integers that minimize variance |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2510.07899 |