Shooting randomly against a line in Euclidean and non-Euclidean spaces

Pub Date : 2014-01-02 DOI:10.1080/17442508.2012.749260
E. Orsingher, Bruno Toaldo
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Abstract

In this paper we study a class of distributions related to the r.v. , for different distributions of . The problem is related to the hitting point of a randomly oriented ray and generalizes the Cauchy distribution in different directions. We show that the distribution of solves the Laplace equation of order , possesses even moments of order , and has bimodal structure when is uniform. We study also a number of distributional properties of functionals of , including those related to the arcsine law. Finally we study the same problem in the Poincaré half-plane and this leads to the hyperbolic distribution of which the main properties are explored. In particular we study the distribution of hyperbolic functions of , the law of sums of i.i.d. r.v.'s and the distribution of the area of random hyperbolic right triangles.
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在欧几里得和非欧几里得空间中对着一条线随机射击
本文研究了一类与rv有关的分布,对于的不同分布。该问题涉及随机定向射线的命中点,并在不同方向上推广柯西分布。证明了该分布解有序拉普拉斯方程,具有偶阶矩,且在均匀时具有双峰结构。我们还研究了泛函的一些分布性质,包括与反正弦律有关的性质。最后,我们在庞卡罗半平面上研究了同样的问题,得到了双曲分布,并探讨了双曲分布的主要性质。特别地,我们研究了双曲函数的分布,i.i.r.v.的和律。和随机双曲直角三角形的面积分布。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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