不等式证明中的概率思想方法

The Probabilistic Thinking Method in Inequality Proving

不等式的证明往往比较复杂,有时直观含义也比较抽象,代数的方法难以发挥作用。如果能够建立适当的概率模型,赋以一些随机事件或随机变量的具体含义,再利用概率的理论加以证明,则常常能使证明过程得到简化。同时还可以为抽象的数学问题提供具体的概率背景,沟通各数学分支之间的联系。文中通过几个不等式的证明阐明了常用的概率思想方法。

Inequality is often more complicated, and sometimes the intuitive meaning is relatively abstract, algebraic methods are difficult to play a role. If we can establish an appropriate probability model, endow specific meaning for some random events or random variables, exert probability theory to prove, then the proof process can often be simplified, can also supply specific probability background for abstract mathematical problem. In this paper, we illustrate probability thinking method through several inequalities proof.