数理统计教学中关于卡方分布的两个问题  

Two Problems on the Chi-Square Distribution in Mathematical Statistics Teaching

作　　者：顾蓓青 张澍轶 王蓉华[2] 徐晓岭 GU Beiqing;ZHANG Shuyi;WANG Ronghua;XU Xiaoling(School of Statistics and Information,Shanghai University of International Business and Economics,Shanghai 201620;Mathematics and Science College,Shanghai Normal University,Shanghai 200234)

机构地区：[1]上海对外经贸大学统计与信息学院,上海201620 [2]上海师范大学数理学院,上海200234

出　　处：《高等数学研究》2025年第1期57-60,共4页Studies in College Mathematics

基　　金：应用统计学国家一流专业建设;上海市高校市级重点课程建设项目“贝叶斯统计”(2024)资助。

摘　　要：针对数理统计教学中关于卡方分布的两个反问题,即若随机变量X=Y^(2)~χ^(2)(1),且Y在(-∞,+∞)上取值,则Y服从N(0,1)是否一定成立(问题一)若随机变量Z与X相互独立,且X=Y^(2)~χ^(2)(1),且Y在(-∞,+∞)上取值,则Z与Y相互独立是否成立(问题二)通过反例说明问题结论不成立,给出了问题成立的条件,并给出了相应的证明过程.This paper addresses two inverse problems related to the chi-square distribution in mathematical statistics teaching.The first problem examines whether a random variable Y necessarily follows a standard normal distribution when a random variable X=Y^(2) followsχ^(2)(1)and Y takes a value on the set(-∞,+∞).The second problem investigates whether the random variables Z and Y are independent when the random variables Z and X are independent,and X=Y^(2) followsχ^(2)(1),and Y takes a value on the set(-∞,+∞).Counterexamples are provided to show that the conclusions of these problems do not hold,and the paper outlines the conditions under which they do,along with the corresponding proof processes.

关 键 词：费歇定理 卡方分布 对称型分布 边际密度 联合密度 

分 类 号：O211[理学—概率论与数理统计]