Some Inequalities Combining Rough and Random Information
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 ra...
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| Format: | Article |
| Language: | English |
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MDPI AG
2018-03-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/20/3/211 |
| _version_ | 1818008374438002688 |
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| author | Yujie Gu Qianyu Zhang Liying Yu |
| author_facet | Yujie Gu Qianyu Zhang Liying Yu |
| author_sort | Yujie Gu |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-14T05:28:27Z |
| format | Article |
| id | doaj.art-9bc3757e9dbb49f093ec85ec721fcab6 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-14T05:28:27Z |
| publishDate | 2018-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-9bc3757e9dbb49f093ec85ec721fcab62022-12-22T02:09:53ZengMDPI AGEntropy1099-43002018-03-0120321110.3390/e20030211e20030211Some Inequalities Combining Rough and Random InformationYujie Gu0Qianyu Zhang1Liying Yu2School of Management, Shanghai University, Shanghai 200444, ChinaSchool of Management, Shanghai University, Shanghai 200444, ChinaSchool of Management, Shanghai University, Shanghai 200444, ChinaRough 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.http://www.mdpi.com/1099-4300/20/3/211rough random variableinequalitiescritical values |
| spellingShingle | Yujie Gu Qianyu Zhang Liying Yu Some Inequalities Combining Rough and Random Information Entropy rough random variable inequalities critical values |
| title | Some Inequalities Combining Rough and Random Information |
| title_full | Some Inequalities Combining Rough and Random Information |
| title_fullStr | Some Inequalities Combining Rough and Random Information |
| title_full_unstemmed | Some Inequalities Combining Rough and Random Information |
| title_short | Some Inequalities Combining Rough and Random Information |
| title_sort | some inequalities combining rough and random information |
| topic | rough random variable inequalities critical values |
| url | http://www.mdpi.com/1099-4300/20/3/211 |
| work_keys_str_mv | AT yujiegu someinequalitiescombiningroughandrandominformation AT qianyuzhang someinequalitiescombiningroughandrandominformation AT liyingyu someinequalitiescombiningroughandrandominformation |