Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (6): 1782-1789.

### 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
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.