[Keyword: Movement] AND [All Categories: Pure Mathematics] : Search

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at customercare@wspc.com for any enquiries.

Advanced Search

SEARCH GUIDE

Results: 1 - 1of1

Follow results:

per page:

Sort: Relevance

Context for search term 1Search term 1*

All Dates

LastSelect static range

Custom Range

Select starting monthSelect starting year

Select ending monthSelect ending year

Search name	Searched On	Run search
[Keyword: Ceramics] AND [All Categories: Geophysics] (3)	27 Mar 2025	Run
[Keyword: Ceramics] AND [All Categories: Chemical Engineering] (10)	27 Mar 2025	Run
[Keyword: Movement] AND [All Categories: Pure Mathematics] (1)	27 Mar 2025	Run
[Keyword: Topology] AND [All Categories: Algebra, Combinatorics, Number Theory &#x0... (1)	27 Mar 2025	Run
[Keyword: Topology] AND [All Categories: Nonlinear Science, Chaos & Dynamical S... (8)	27 Mar 2025	Run

articleNo Access
Asymptotic formula for sum of moment mean deviation for order statistics from uniform distribution
- Rafał Kapelko
Discrete Mathematics, Algorithms and Applications01 Apr 2019
Preview Abstract
Assume that $n$ $n$ mobile sensors are thrown uniformly and independently at random with the uniform distribution on the unit interval. We study the expected sum over all sensors $i$ $i$ from $1$ $1$ to $n,$ $n,$ where the contribution of the $i^{th}$ $i^{th}$ sensor is its displacement from the current location to the anchor equidistant point $t_{i} = \frac{i}{n} - \frac{1}{2 n},$ $t_{i} = \frac{i}{n} - \frac{1}{2 n},$ raised to the $a^{th}$ $a^{th}$ power, when $a$ $a$ is an odd natural number.
As a consequence, we derive the following asymptotic identity. Fix $a$ $a$ positive integer. Let $X_{i : n}$ $X_{i : n}$ denote the $i^{th}$ $i^{th}$ order statistic from a random sample of size $n$ $n$ from the Uniform $(0, 1)$ $(0, 1)$ population. Then
$n \sum i = 1 E [∣ ∣ ∣ ∣ X_{i : n} - E [X_{i : n}] {∣ ∣ ∣ ∣}^{a}] = \frac{Γ (\frac{a}{2} + 1)}{2^{\frac{a}{2}} (1 + a)} \frac{n}{n^{\frac{a}{2}}} + O (\frac{\sqrt{n}}{n^{\frac{a}{2}}}),$ $\sum_{i = 1}^{n} E [| X_{i : n} - E [X_{i : n}] |^{a}] = \frac{Γ (\frac{a}{2} + 1)}{2^{\frac{a}{2}} (1 + a)} \frac{n}{n^{\frac{a}{2}}} + O (\frac{\sqrt{n}}{n^{\frac{a}{2}}}),$
where $Γ (z)$ $Γ (z)$ is the Gamma function.

1