随机变量函数分布的求解方法再探索

Reexplore the Solution Method of the Function Distribution of Random Variables

摘要: 国内现行概率论教材中，随机变量函数的分布是重要知识点，但分布较杂、较散，不利于高校师生系统教学与学习。基于此，本文归纳了各类常见的随机变量函数的分布，并针对应用最广泛的二维连续型随机变量函数分布进行优化拓展，使问题易于理解，计算更为方便。主要表现在：1) 利用一维数轴，降维处理变量取值范围问题；2) 引入一个重要定理，将各类二维连续型随机变量函数分布的求解一般化；3) 引入国外教材“总–分–总”的解题思路，拓宽新的解题路径；4) 简述多维随机变量函数分布的解题技巧。最终给出合理解题路径选择。

Abstract: In the current domestic probability theory textbooks, the distribution of random variable functions is an important knowledge point, but the distribution is more mixed and scattered, which is not conducive to the systematic teaching and learning of teachers and students in colleges and universities. Based on this, this paper summarizes the distribution of various common random variable functions, and optimizes and expands the distribution of functions of the most widely used two-dimensional continuous random variable, making the problem easy to understand and more convenient to calculate. It is mainly manifested in: 1) Using the one-dimensional number axis to reduce the dimensionality to deal with the variable value range problem; 2) Introduce an important theorem to generalize the solution of various two-dimensional continuous random varia-ble function distributions; 3) Introduce the solution ideas of foreign textbooks “total-score-total” to broaden the new solution path; 4) Briefly describe the problem solving techniques of multidimensional random variable function distribution. Finally, the path selection of the comprehension question is given.