切比雪夫不等式及其应用(摘要)

切比雪夫不等式及其应用

摘要

切比雪夫不等式是概率论中重要的不等式之一。尤其在分布未知时，估计某些事件的概率的上下界时，常用到切比雪夫不等式。另外，大数定律是概率论极限理论的基础，而切比雪夫不等式又是证明大数定律的重要途径。如今，在切比雪夫不等式的基础上发展起来的一系列不等式都是研究中心极限定理的有力工具。作为一个理论工具，切比雪夫不等式的地位是很高的。

本文首先介绍了切比雪夫不等式的一些基本理论，引出其概率形式，用现代概率方法证明了切比雪夫不等式并给出了其等号成立的充要条件。其次，从三大方面阐述了其在概率论中的应用，并且给出了切比雪夫大数定律和伯努利大数定律的证明。在充分了解切比雪夫不等式后，最后探索了其在生活中的应用，并且用切比雪夫不等式评价了IRR的概率风险分析。

关键词:切比雪夫不等式 大数定律 IRR

The Chebyster’s Inequality and Its Applications

ABSTRACT

In probability theory, the Chebyshev’s Inequality is one of the important inequalities. In particular the distribution is unknown, the Chebyshev’s Inequality is usually used when estimating the boundary from above or below of probability. In addition, the Law Of Large Numbers is the basis of the limit theory of probability. The Chebyshev’s Inequality is an important way to prove it. Now, a series of inequalities that are developed on the basis of the Chebyshev’s Inequality are a powerful tool for the Central Limit Theorem. As a theoretical tool, its status is very high.

First, this article introduces some basic theory of the Chebyshev’s Inequality, it raises the Chebyshev’s Inequality’s form of probability and makes a prove for the Chebyshev’s Inequality with the method of modern probability. Furthermore, it gives the necessary and sufficient condition of the establishment of the equal sign.

Secondly, we introduces its five application in probability theory and gives the prove of the Chebyshev and Bernoulli Law Of Large Numbers. After the full understanding of the Chebyshev’s Inequality, finally, we explore its application in the life and give the probabilistic risk assessment of the IRR with the Chebyshev’s Inequality.

Key Words：Chebyshev’s Inequality Law Of Large Numbers IRR