Demystifying the Integrated Tail Probability Expectation Formula
Demystifying the Integrated Tail Probability Expectation Formula
复制标题
揭秘综合尾部概率期望公式
DOI：
10.1080/00031305.2018.1497541
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Ambrose Lo
中科院分区：
文献类型：
--
作者：
Ambrose Lo
计算不同类型随机变量的期望值是数理统计中的一个中心问题。针对学生和教师在介绍概率和统计课程和研究生水平的测度论概率课程，这教学笔记投光一般期望公式表示的分布和生存函数的随机变量，并讨论其教育价值。这个非传统的期望公式通常被放在数理统计教科书的章节结尾练习中，很少讨论，并在多余的技术假设下提出，它为学生提供了一个宝贵的机会来欣赏期望的几何意义，这在大多数本科生和研究生课程中被忽视，并且用作用于计算期望值的有效工具，而通过传统手段可能要费力得多。为了学生的利益，这个公式值得在课堂上结合期望的教学进行彻底的处理。除了澄清一些常见的误解和显示的教学价值的期望公式，这说明提供指导教师在教学公式考虑到目标学生群体的背景。
ABSTRACT Calculating the expected values of different types of random variables is a central topic in mathematical statistics. Targeted toward students and instructors in both introductory probability and statistics courses and graduate-level measure-theoretic probability courses, this pedagogical note casts light on a general expectation formula stated in terms of distribution and survival functions of random variables and discusses its educational merits. Often consigned to an end-of-chapter exercise in mathematical statistics textbooks with minimal discussion and presented under superfluous technical assumptions, this unconventional expectation formula provides an invaluable opportunity for students to appreciate the geometric meaning of expectations, which is overlooked in most undergraduate and graduate curricula, and serves as an efficient tool for the calculation of expected values that could be much more laborious by traditional means. For students’ benefit, this formula deserves a thorough in-class treatment in conjunction with the teaching of expectations. Besides clarifying some commonly held misconceptions and showing the pedagogical value of the expectation formula, this note offers guidance for instructors on teaching the formula taking the background of the target student group into account.