【论文】大数定律及其应用.pdf

大数定律及其应用 摘 要 本文通过对概率论的经典定理—大数定律在独立同分布和不同分布两种情况 下的结论作了比较系统的阐述，揭示了随机现象最根本的性质—平均结果的稳定 性。较全面地分析和叙述了大数定律在独立不同分布条件下的马尔科夫定理以及作 为它的推论的切比雪夫定理、伯努利定理、泊松定理和同分布条件下的辛钦定理， 而且把这些定理推广到以概率1 收敛意义下的强大数定律。另外，叙述了各种大数 定律之间的关系。同时通过举出很多相关的正反例子，进行说明这些定理所给出的 条件是否是充要条件，来强调在实际问题中灵活地应用和辨别是否服从我们给出的 定理条件。最后给出了一些简便的大数定律在数理统计、近似计算、以及保险业等 方面的应用，来进一步地阐明了大数定律在各分支学科中的重要作用和应用价值。 关键词：随机变量，随机变量的收敛，独立随机变量，大数定律 I 大数定律及其应用 Abstract This paper by means of the studying on laws of large numbers under the condition either of independent and identically distributions or independent but different distributions comprehensively, we illuminate that the mean value is stable which is the essential property for the stochastic event. At the same time, we analyzed and stated the Markov theorem under the different terms and conditions, so its deductive theorems, i.e. Chebyshev theorem, Bernoulli theorem, Poisson theorem including Khintchine theorem based on the identically distribution. Moreover, we have got the strong laws of large numbers in the sense of its probability with 1 by generalization. In addition, we showed the relations between various laws of large numbers. The more interesting is that plenty of correlative examples including positive and negative are illustrated to emphasize the importance of identifying the conditions of these theorems in all sorts of their applications. At last, we brought forth the value of laws of large numbers in some subjects such as mathematical statistics, approximate calculation, insurance and so on. Keywords: random variable, convergence of random variable, independent random variables, laws of large number