非负随机变量函数的期望的另一公式及其应用  

Another formula for the expectation of a function of a random variable and its application

作　　者：王霞[1] 马晓雯 WANG Xia;MA Xiaowen(School of Mathematics,Statistics and Mechanics,Beijing University of Technology,Beijing 100124,China;School of Mathematics,Qilu Normal University,Jinan 250200,China)

机构地区：[1]北京工业大学数学统计学与力学学院,北京100124 [2]齐鲁师范学院数学学院,山东济南250200

出　　处：《南阳师范学院学报》2025年第1期35-37,共3页Journal of Nanyang Normal University

基　　金：国家自然科学基金项目(62072044)。

摘　　要：探讨了随机变量函数的数学期望问题,特别是考虑了随机变量非负且函数在非负实数区间上严格单调递增的正的一阶可导函数的情况。提出了一个新的公式来计算这类函数的数学期望,该公式的积分表达式与非负随机变量的尾概率和函数的导数有关。通过Fubini定理和变量替换方法给出了两个计算期望的公式,然后提供了两个应用实例,通过实例展示了如何将定理应用于具体的函数。In this paper,we discuss the issue of mathematical expectation of functions of random variables,especially when the random variables are non-negative and the functions are strictly monotonically increasing positive first-order differentiable functions on the non-negative real number interval.The article proposes a new formula for calculating the mathematical expectation of such functions,whose integral expression is related to the tail probability of the non-negative random variable and the derivative of the function.The article first presents two formulas for calculating expectations by utilizing Fubini's theorem and variable substitution methods.Then,two application examples are provided.These applications demonstrate how to apply the theorem to specific functions.

关 键 词：非负随机变量 随机变量的函数 数学期望 

分 类 号：O212.1[理学—概率论与数理统计]