Fuzzy Optimization and Decision Making

, Volume 14, Issue 1, pp 57–76 | Cite as

Some results of moments of uncertain variable through inverse uncertainty distribution

  • Yuhong Sheng
  • Samarjit KarEmail author


Uncertainty theory is a branch of axiomatic mathematics for modeling belief degrees. This theory provides a new mathematical tool for indeterminacy phenomena. Uncertain variable is a fundamental concept in uncertainty theory and used to represent quantities with uncertainty. There are some important characteristics about uncertain variables. The expected value of uncertain variable is its average value in the sense of uncertain measure and represents the size of uncertain variable. The variance of uncertain variable provides a degree of the spread of the distribution around its expected value. The moments are some other important characteristics of an uncertain variable. In order to describe the moment of uncertain variable, this paper provides a new formula using inverse uncertainty distribution. Several practical examples are provided to calculate the moments using inverse uncertainty distribution. In addition, some new formulas are derived to calculate the central moments of uncertain variables through uncertainty distribution and inverse uncertainty distribution.


Uncertainty theory Uncertain variable Inverse uncertainty distribution Moments 



This work was supported by National Natural Science Foundation of China Grants Nos. 61273044 and 91224008.


  1. Chen, X. W., & Dai, W. (2011). Maximum entropy principle for uncertain variables. International Journal of Fuzzy Systems, 13(3), 232–236.MathSciNetGoogle Scholar
  2. Chen, X. W., & Kar, S. (2012). Cross entropy measure of uncertain variables. Information Science, 201, 53–60.CrossRefzbMATHMathSciNetGoogle Scholar
  3. Dai, W., & Chen, X. W. (2012). Entropy of function of uncertain variables. Mathematical and Computer Modelling, 55(3–4), 754–760.CrossRefzbMATHMathSciNetGoogle Scholar
  4. Gao, X. (2009). Some properties of continuous uncertain measure. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 17(3), 419–426.CrossRefzbMATHMathSciNetGoogle Scholar
  5. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–292.CrossRefzbMATHGoogle Scholar
  6. Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Julius Springer.CrossRefGoogle Scholar
  7. Liu, B. (2007). Uncertainty theory (2nd ed.). Berlin: Springer.zbMATHGoogle Scholar
  8. Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3–10.Google Scholar
  9. Liu, B. (2010). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer.CrossRefGoogle Scholar
  10. Liu, B. (2012). Why is there a need for uncertainty theory? Journal of Uncertain Systems, 6(1), 3–10.Google Scholar
  11. Liu, B. (2013). Toward uncertain finance theory. Journal of Uncertainty Analysis and Applications, 1(1), 1–15.CrossRefGoogle Scholar
  12. Liu, Y. H., & Ha, M. H. (2010). Expected value of function of uncertain variables. Journal of Uncertain Systems, 4(3), 181–186.Google Scholar
  13. Peng, Z. X., & Iwamura, K. (2010). A sufficient and necessary condition of uncertainty distribution. Journal of Interdisciplinary Mathematics, 13(3), 277–285.CrossRefzbMATHMathSciNetGoogle Scholar
  14. Wang, X. S. & Peng, Z. X. (2014). Method of moments for extimating uncertainty distributions. Journal of Uncertainty Analysis and Applications, 2(5). doi: 10.1186/2195-5468-2-5.
  15. Yao, K. (2013). A formula to calculate the variance of uncertain variable.

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  2. 2.College of Mathematical and System SciencesXinjiang UniversityUrumchiChina
  3. 3.Department of MathematicsNational Institute of Technology DurgapurDurgapurIndia

Personalised recommendations