概率方法证明不等式

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翻新时间：2023-04-07

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概率方法证明不等式

摘要

在分析数学中有些不等式的证明往往比较复杂，若运用代数方法较难得到解决，而且具体的直观含义也比较抽象。因此，为了简化这些不等式的证明过程，通过阅读大量的相关资料，本文从数学的基本概念入手，运用了1种巧妙的方法——概率方法，即根据不等式的主要特征，结合概率论的1些基本概念和公式，通过建立1个适当的概率模型，赋以1些随机事件或随机变量的具体含义，再利用概率论的理论加以证明，讨论了柯西（Cauchy）不等式，级数不等式，詹森(Jensen)不等式和几个1般不等式的证明。

本文通过对利用概率方法证明不等式进行了分析，从而得出：概率方法可以为抽象的数学问题提供具体的概率背景，同时还沟通了各数学分支之间的联系；显示了概率应用的巧妙性和优越性，大大简化了不等式的证明过程。

关键词：随机变量；概率方法；数学不等式

Abstract

In analysis mathematics，some identifications of inequalities are often more complicated，It will be difficult to prove that if using algebra method，and concrete ocular meaning is more abstract．Therefore，in order to simplify the proving process of these inequalities．Though reading a lot of relevant resource，we begin with the basic concept of math，and use an ingenious way――probabilistic method, which means that according to the main features of inequality theory，combining the basic concepts and formulas of probability，through creating one suitable probability model，giving some concrete meanings of random events or random variables，proving through probability theory，we discuss the Cauchy inequality，Class inequality，Jensen inequality，and several common inequality’s proofs．We analyze the inequalities which is proved by probabilistic method，and at last we obtain that probability method can provide a concrete probability background for abstract mathematics branch，and greatly simplify the proving process of inequalities．

Keywords: random variable; probability methob; mathematical inequalities