arXiv
Open Access
2016
On the Moment Distance Between Sensors and Anchor Points
Rafał Kapelko
Abstrak
The present paper contains additional asymptotic result over an earlier investigation of Kapelko and Kranakis. Consider $n$ mobile sensors placed independently at random with the uniform distribution on the unit interval $[0,1]$. Fix $a$ an odd natural number. Let $X_i$ be the the $i-$th closest sensor to $0$ on the interval $[0,1].$ Then the following identity holds $$\sum_{i=1}^n\mathbf{E}\left[\left|X_i-\left(\frac{i}{n}-\frac{1}{2n}\right)\right|^a\right]=\frac{Γ\left(\frac{a}{2}+1\right)}{2^{\frac{a}{2}}(1+a)}\frac{1}{n^{\frac{a}{2}-1}}+O\left(\frac{1}{n^{\frac{a-1}{2}}}\right),$$ when $a$ is an odd natural number, where $Γ(z)$ is the Gamma function.
Topik & Kata Kunci
Penulis (1)
R
Rafał Kapelko
Akses Cepat
Informasi Jurnal
- Tahun Terbit
- 2016
- Bahasa
- en
- Sumber Database
- arXiv
- Akses
- Open Access ✓