摘要:对于等式和不等式的证明，人们通常是采用构造法、作差法或者是归纳法等方法，但是在分析数学中有些等式和不等式的证明通常比较复杂，若运用常见的代数方法难以得到解决，并且直观的含义也比较抽象。而本文主要通过构造适当的概率模型的方法，结合概率论的一些基本概念和公式，赋以一些随机变量的具体含义，再运用概率论的理论加以证明，使得一些原本很难证明的等式和不等式变得简单而容易。

关键词：等式证明; 不等式证明; 概率论

Abstract：Usually people use a constructive method, difference method or induction method to prove the equalities and inequalities, but the analysis of some equalities and inequalities proof in mathematics are usually more complex and difficult to be solved if with algebraic methods, and the intuitive meaning is more abstract. In this paper, the method of constructing a suitable probability model, combined with some basic concepts of probability theory and formulas, endowed with some of the special meaning of the random variable, then using the theory of probability theory to prove, which can make some very difficult to prove equalities and inequality has become simple and easy.

Key words: Proof of equality;  Proof of inequality;  Probability

　　证明等式与不等式比较常见的方法有比较法、综合法、分析法、反证法、放缩法、构造法和数学归纳法等。这些方法它们各有千秋，而其中比较法是证明等式和不等式的最基本的方法，它思维清晰，可操作性强，适用范围广泛，在证明中常常采用。但是在有的问题情境中比较法乃至其它方法并不能够和轻易地就证得结论，我们往往需要通过别的比较特殊的方法加以证明。而概率论不仅仅是用来算算概率的，有些时候概率论往往比我们想象中的还要强大。