数学物理学报 ›› 2022, Vol. 42 ›› Issue (6): 1782-1789.

• 论文 • 上一篇    

独立同分布随机变量加权和的概率估计

马丽1,2, 叶柳2   

  1. 1. 海南师范大学数据科学与智慧教育教育部重点实验室, 海口 571158;
    2. 海南师范大学数学与统计学院, 海口 571158
  • 收稿日期:2021-08-05 修回日期:2022-04-29 发布日期:2022-12-16
  • 通讯作者: 叶柳,E-mail:1187621908@qq.com E-mail:1187621908@qq.com
  • 作者简介:马丽,E-mail:malihnsd@163.com
  • 基金资助:
    国家自然科学基金$(11861029)、海南省自然科学基金(122MS056,120RC589)和海南省研究生创新科研课题(Ohys2021-301)

Probability Estimation of the Weighted Sum of Independent Identically Distributed Random Variables

Ma Li1,2, Ye Liu2   

  1. 1. Key Laboratory of Data Science and Smart Education, Ministry of Education, Hainan Normal University, Haikou 571158;
    2. Department of Mathematics and Statistic, Hainan Normal University, Haikou 571158
  • Received:2021-08-05 Revised:2022-04-29 Published:2022-12-16
  • Supported by:
    Supported by the NSFC(11861029), the Hainan Provincial Natural Science Foundation(122MS056, 120RC589) and the Hainan Postgraduate Innovative Research Project(Ohys2021-301)

摘要: 设$\{\xi_{i}\}_{i=1}^n$为独立同分布的随机变量,且$P(\xi_i=1)=P(\xi_i=-1)=\frac{1}{2}$.设$\overrightarrow{a}=(a_{1},\cdots,a_{n})$为与$\{\xi_{i}\}_{i=1}^n$独立的服从超球面$S^{n-1}=\{(a_{1},\cdots,a_{n})\in\mathbb{R}^n|\sum\limits^n_{i=1}a_i^2=1\}$上均匀分布的随机变量,该文用极坐标变换得到了$P(|\sum\limits_{i=1}^n{a_i}{\xi_{i}|\leq1})$的表达式.当$n\leq7$时,该文通过直接计算得到此概率值大于等于$\frac{1}{2}$;当$n\geq8$时,该文通过R软件也得到了此概率值大于等于$\frac{1}{2}$.特别地,$\!n=3,4$时,借助于贝塔函数,该文直接证明了该概率值大于等于$\frac{1}{2}$.

关键词: 独立同分布随机变量, 加权和, 概率估计

Abstract: Let $\xi_{i}(1\leq{i}\leq{n})$ be independent identically distributed random variables satisfying $P(\xi_i=1)=P(\xi_i=-1)=\frac{1}{2}$. Let $\overrightarrow{a}=(a_{1},\cdots,a_{n})$ be random variables uniformly distributed on $S^{n-1}=\{(a_{1},\cdots,a_{n})\in\mathbb{R} ^n|\sum\limits^n_{i=1}a_i^2=1\}$ which are independent of $\xi_{i}(1\leq{i}\leq{n})$. In this paper, we get the expression of $P (|\sum\limits_{i=1}^n{a_i}{\xi_{i}|\leq1})$ by polar coordination transformation. For $n\leq7$, we give the value of $P (|\sum\limits_{i=1}^n{a_i}{\xi_{i}|\leq1})$ directly which is no less than one half. For $n\geq8$, we can use R software to calculate the value which is also no less than one half. Moreover, for $n=3,4$, by Beta function, we show that the probability value is still no less than one half.

Key words: Independent identically distributed random variable, Weighted sum, Probability estimation

中图分类号: 

  • O211.4