协方差不等式的若干证明及其注记——兼论大学数学各学科教学的融合  

Proofs of Covariance Inequality with a Note on the Interdisciplinary Teaching of College Mathematics

作　　者：欧阳顺湘 OUYANG Shunxiang(School of Science,Harbin Institute of Technology(Shenzhen),Shenzhen Guangdong 518055,China)

机构地区：[1]哈尔滨工业大学(深圳)理学院,广东深圳518055

出　　处：《大学数学》2022年第5期81-88,共8页College Mathematics

摘　　要：介绍协方差不等式的十种证明并指出证明之间的一些联系.证明分别用到数学中的归一化、不相关化、构造辅助函数、投影、恒等式、求极值等常见思想和方法,还涉及商空间、内积空间等概念.这些证明充分展示了分析、代数、几何与概率论及数理统计之间的密切联系.由此强调习惯上分门别类进行的大学数学各学科的教学应当相互融合,以使学生感知到数学是一个有机整体.We present ten proofs of covariance inequality by applying various classical methods such as normalization,making random variables uncorrelated,construction of functions,construction of identity,projection and minimization.Concepts like quotient space and inner space etc.are also used.It shows that there are close connections among analysis,algebra,geometry,probability theory and mathematical statistics.Interdisciplinary teaching of college mathematics courses is hence emphasized to show the unity of college mathematics to students.

关 键 词：协方差不等式 柯西-施瓦茨不等式 大学数学 教学融合 方差 协方差 相关系数 

分 类 号：O211.5[理学—概率论与数理统计]