Gu Yujie, Zhang Qianyu, Yu Liying
School of Management, Shanghai University, Shanghai 200444, China.
Entropy (Basel). 2018 Mar 20;20(3):211. doi: 10.3390/e20030211.
Rough random theory, generally applied to statistics, decision-making, and so on, is an extension of rough set theory and probability theory, in which a rough random variable is described as a random variable taking "rough variable" values. In order to extend and enrich the research area of rough random theory, in this paper, the well-known probabilistic inequalities (Markov inequality, Chebyshev inequality, Holder's inequality, Minkowski inequality and Jensen's inequality) are proven for rough random variables, which gives a firm theoretical support to the further development of rough random theory. Besides, considering that the critical values always act as a vital tool in engineering, science and other application fields, some significant properties of the critical values of rough random variables involving the continuity and the monotonicity are investigated deeply to provide a novel analytical approach for dealing with the rough random optimization problems.
粗糙随机理论一般应用于统计学、决策等领域，是粗糙集理论和概率论的一种扩展，其中粗糙随机变量被描述为取值为“粗糙变量”的随机变量。为了扩展和丰富粗糙随机理论的研究领域，本文针对粗糙随机变量证明了著名的概率不等式（马尔可夫不等式、切比雪夫不等式、赫尔德不等式、闵可夫斯基不等式和詹森不等式），这为粗糙随机理论的进一步发展提供了坚实的理论支持。此外，鉴于临界值在工程、科学和其他应用领域中始终是一种重要工具，本文深入研究了粗糙随机变量临界值的一些重要性质，包括连续性和单调性，以便为处理粗糙随机优化问题提供一种新颖的分析方法。
