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