方差相关恒等式与不等式——兼谈教学中的“见微知著”与“举重若轻”  

On Variance Related Identities and Inequalities with a Discussion of Strategies for Teaching Simple Facts and Difficult Subjects

作　　者：欧阳顺湘 OUYANG Shunxiang(School of Electrics and Computer Engineering,Nanfang College Guangzhou,Guangzhou 510970,China)

机构地区：[1]广州南方学院电气与计算机工程学院,广州510970

出　　处：《大学数学》2024年第5期85-99,共15页College Mathematics

摘　　要：从两个基本的方差恒等式出发,讨论与方差相关的一些恒等式与不等式及其应用.主要内容包括方差的非显式地依赖于期望的表示,切比雪夫序不等式(FKG不等式),用方差衡量延森不等式两边的亏损(Φ熵不等式),Bregman散度与期望的联系,由方差相关等式得到的惠更斯-莱布尼兹恒等式、第二拉格朗日恒等式等代数恒等式及其在统计学教学中的应用.最后结合本文内容简要讨论教学中应该注意“见微知著”式的启发,“举重若轻”和“删繁就简”的教学方法,以及数学各学科的融合.We prove an equality without using expectation explicitly for the variance of a random variable by using identical independent random variables.Chebyshev’s order inequality are also introduced to emphasize this probabilistic method.Motivated by two fundamental identities of variance,we show some results on the defects of Jensen’s inequalities via variance.Bregman’s divergence and the role of expectation are introduced.Some algebraic identities,e.g.Huygens-Leibniz’s identity and the second Lagrange’s identity,are obtained as consequences of the two fundamental identities of variance.Applications of these algebraic identities in the teaching of mathematical statistics are also presented.Finally we present a discussion of strategies for teaching simple facts and difficult subjects.

关 键 词：方差 延森不等式 Φ熵 Bregman散度 惠更斯-莱布尼兹恒等式 第二拉格朗日恒等式 切比雪夫序不等式 

分 类 号：O211.5[理学—概率论与数理统计]