独立同分布随机变量加权和的概率估计

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

设 {ξ_(i(}^(n)_(i=1)为独立同分布的随机变量,且P(ξ_(i)=1)=P(ξ_(i)=−1)=1/2.设a→=(a_(1),…,a_(n))为与{ξ_(i)}_(i=1)^(n)独立的服从超球面S^(n−1)={(a_(1),…,a_(n))∈R^(n)|∑_(i=1)^(n)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(|∑_(i=1)^(n)a_(i)ξ_(i)|≤1)的表达式.当n≤7时,该文通过直接计算得到此概率值大于等于1/2;当n≥8时,该文通过R软件也得到了此概率值大于等于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1/2.

Let ξ_(i)(1≤i≤n)be independent identically distributed random variables satisfyingP(ξ_((i)=1)=P(ξ_(i)=−1)=1/2.Let a→=(a_(1),…,a_(n))be random variables uniformly distributed on S^(n−1)={(a_(1),…,a_(n))∈R^(n)|∑_(i=1)^(n)a_(i)^(2)=1} which are independent ofξ_(i)(1≤i≤n).In this paper,we get the expression of P(|∑_(i=1)^(n)a_(i)ξ_(i)|≤1)by polar coordination transformation.For n≤7,we give the value of P(|∑_(i=1)^(n)a_(i)ξ_(i)|≤1)directly which is no less than one half.For n≥8,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.