概率论方法在数学分析中的一些应用

概率论方法在数学分析中的一些应用Some applications of probability theory in the mathematical

analysis

摘要

概率论作为数学的一个分支，与其他学科分支有着密切的联系，具有广泛的应用性。著名的数学家王梓坤院士在文献中[1]指出：“用概率论的方法来证明一些关系式或者解决其他数学分析中的问题，是概率论的重要研究方向之一。”概率论方法不仅能解决一些随机的数学问题，而且还可以解决一些确定的数学问题[2]，而且某些在数学分析中很难解决的问题，只要运用合适的概率论模型或是定理，就能得到很好的解决。然而现如今多数有关概率论与数学分析联系的文献不是很全面，本文归纳概括了概率论在数学分析中的应用，选择了比较典型的五类：概率论方法解决极限问题、概率论方法解决无穷级数问题、概率论方法解决积分问题、概率论方法解决恒等式问题及概率论方法解决不等式问题，而且在每一类的问题讨论中引入很多概率论中的定理和公式，清晰地阐述概率论在数学分析知识间的运用。

Probability theory as a branch of mathematics has close connection with other subjects and their branches, it has wide applicability. Famous mathematician and academician Wong Chi-Kun, in his literature[1] pointed out: "It is one of the most important research directions to use probability theory to prove some relationship or to solve the problems in mathematical analysis.” Probability theory can not only solve some random math problems, but also can solve some identified mathematical problems[2], What’s more some very difficult questions in mathematical analysis can also be well resolved by using a suitable probability theory model or theorem. However, now most literatures which related to the relationship of probability theory and mathematical analysis are not very comprehensive, This article summarized the outlines of probability theory in mathematical analysis, selected five typical themes: probability theory in solving the ultimate problem, probability theory in solving the problem of infinite series, probability theory in solving the integral problem, probability theory in solving the identity problem and probability theory in solving the inequality problem. Furthermore in

order to represent a clear idea on the use of probability theory in Mathematical Analysis, this article introduced a lot of theorems and formulas related to probability theory when discussing every theme.

关键词：贝努利模型；正态分布；泊松分布；中心极限定理；大数定理；Cauchy-Schwartz 不等式；随机变量

Keyword: Bernoulli model; normal distribution; poisson distribution; central limit theorem; law of large numbers; Cauchy-Schwartz inequality; random variable

目录

引言 (4)

一、概率论方法解决极限问题 (5)

（一）概率论方法解决极限问题概述 (5)

（二）典型例题分析与证明 (5)

（三）概率论方法解决极限问题的意义 (8)

二、概率论方法解决无穷级数问题 (9)

（一）概率论方法解决无穷级数问题概述 (9)

（二）典型例题分析与证明 (9)

（三）概率论方法解决无穷级数问题的意义 (14)

三、概率论方法解决积分问题 (15)

（一）概率论方法解决积分问题概述 (15)

（二）典型例题分析与证明 (15)

（三）概率论方法解决积分问题的意义 (19)

四、概率论方法解决恒等式问题 (19)

（一）概率论方法解决恒等式问题概述 (19)

（二）典型例题分析与证明 (19)

（三）概率论方法解决恒等式问题的意义 (21)

五、概率论方法解决不等式问题 (21)

（一）概率论方法解决不等式问题概述 (21)

（二）典型例题分析与证明 (21)

（三）概率论方法解决不等式问题的意义 (26)

结论 (26)

参考文献 (27)

致谢 (28)