概率论中几个不等式的推广及应用

The Generalizations and Applications of Several Inequalities in Probabilistic Theory

Markov不等式和Chebyshev不等式是概率论中两个重要的不等式。从这两个不等式和中心极限定理出发,利用指数函数y=esx,s>0的单调增加和凹性得到了几个新的不等式。并对服从两点分布的独立随机变量X1,X2,…,Xn进行研究,首先利用Chebyshev不等式得到随机变量Sn=∑ni=1Xi偏离方差ESn的上界p(1-p)/nε2,但它只有O〔1/n〕阶的收敛速度,然后利用新的不等式得到一个新的上界e-2nε2,它有更快的收敛速度,这在探讨收敛速度时有着重要的理论意义。

Markov＇s inequality and Chebyshev＇s inequality are very important in probabilistic theory. From the two inequalities and the central limit theorem, some new inequalities are derived by using the properties of monotonous increase and concave of the function, y=e^sx, s〉O, Assume that Xl ,X2 ,…… ,Xn, are independent identically distributed （i. i. d. ） Bernoulli（p） random variables. First, we can get the upper bound p（1-p）/nε^2 with order O（1/n） from Chebyshev inequality. Second, a better upper bound e^-2m^2 was gotten with the new inequalities, which can fasten the convergence rate.