论文标题
BienayMé-Chebyshev和Gaußtype的急剧不平等可能在平均值周围不对称间隔
Sharp Inequalities of Bienaymé-Chebyshev and GaußType for Possibly Asymmetric Intervals around the Mean
论文作者
论文摘要
Gauß（1823）证明，当随机变量分布在零时，模式为零时，随机变量落在零附近的对称间隔之外的概率上证明了一个尖锐的上限。对于所有分布的类别，平均值为零，Bienaymé（1853）和Chebyshev（1867）独立地提供了另一个概率上的更简单的尖锐上限。对于相同类别的分布，Cantelli（1928）获得了一个严格的上限，该间隔为半线。我们将这些结果扩展到六类分布的任意间隔，即“分布”的一般类别，“凹形分布”的类别的“分布”的类别，“非模态分布”，``nun -opodal分布''，``单峰分布''，具有巧合模式和均值模式和``对称的单模式分布''的类别。对于某些已知的不平等，例如Gauß\，不等式，给出了另一种证据。
Gauß(1823) proved a sharp upper bound on the probability that a random variable falls outside a symmetric interval around zero when its distribution is unimodal with mode at zero. For the class of all distributions with mean at zero, Bienaymé (1853) and Chebyshev (1867) independently provided another, simpler sharp upper bound on this probability. For the same class of distributions, Cantelli (1928) obtained a strict upper bound for intervals that are a half line. We extend these results to arbitrary intervals for six classes of distributions, namely the general class of `distributions', the class of `symmetric distributions', of `concave distributions', of `unimodal distributions', of `unimodal distributions with coinciding mode and mean', and of `symmetric unimodal distributions'. For some of the known inequalities, such as the Gauß\, inequality, an alternative proof is given.
