样本容量为2的标准柯西分布次序统计量的性质研究  

On Properties of Order Statistics for Standard Cauchy Distribution with Sample Size 2

作　　者：邱珍珠 尹真真 QIU Zhenzhu;YIN Zhenzhen(School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai 201620)

机构地区：[1]上海对外经贸大学统计与信息学院,上海201620

出　　处：《高等数学研究》2022年第3期34-37,共4页Studies in College Mathematics

摘　　要：设X服从标准柯西分布(X~C(1)),X_(1),X_(2)是取自总体X的简单随机样本,其次序统计量记为X_((1))=min(X_(1),X_(2)),X_((2))=max(X_(1),X_(2)).本文给出了X_((1)),X_((2))的分布函数和联合分布函数,X_(2)(1),X_(2)(2)的分布函数和联合分布函数,得到了如下一些结论:X_((1))与X_((2))既不独立也不同分布;X_((1))与-X_((2))同分布,X_((2))与-X_((1))同分布X_((1))与X_((2))独立同分布X_(2)(1)与X_(2)(2)独立同分布于F(1,1)分布X_(1)/X_(2),X_(2)/X_(1)与X_((1))/X_((2)),X_((2))/X_((1))同分布X_(2)1/X_(2)2,X_(2)2/X_(2)1与X_(2)(1)/X_(2)(2),X_(2)(2)/X_(2)(1)同分布.Let X_(1),X_(2) be a simple random sample taken from the population X which follows the standard Cauchy distribution(X~C(1)).Denote the order statistics by X_((1))=min(X_(1),X_(2)),X_((2))=max(X_(1),X_(2)).In this paper,the distribution function and the joint distribution function of X_((1)),X_((2))and X_(2)(1),X_(2)(2)are presented.The following conclusions are obtained:X_((1))and X_((2))are neither independent nor distributed identically;X_((1))and-;X_((2))have the same distribution,X_((2))and - X_((1))have the same distribution;X_((1))and X_((2))are independent and identically distributedX_(2)(1)and X_(2)(2)are independent and identically distributed with F(1,1)X_(1)/X_(2),X_(2)/X_(1)and X_((1))/X_((2)),X_((2))/X_((1))have the same distribution;X_(1)^(2)/X_(2)^(2),X_(1)^(2)/X_(2)1 and X_(2)(1)/X_(2)(2),X_(2)(2)/X_(2)(1)have the same distribution.

关 键 词：标准柯西分布 次序统计量 分布函数 

分 类 号：O211[理学—概率论与数理统计]