Document Type : Research Paper


1 Department of Statistics, University of Sistan and Baluchestan, Zahedan, Iran

2 College of Mathematical and System Sciences, Xinjiang University, Urumqi 830046, China


Chance theory is a mathematical methodology for dealing with indeterminate
phenomena including uncertainty and randomness.
Consequently, uncertain random variable is developed to describe the phenomena which involve
uncertainty and randomness.
Thus, uncertain random variable is a fundamental concept in chance theory.
This paper provides some practical quantities to describe uncertain random variable.
The typical one is the expected value, which is the uncertain version of the
center of gravity of a physical body.
Mathematically, expectations are integrals with respect to chance distributions
or chance measures.
In fact, expected values measure the center of gravity of a distribution; they are
measures of location. In order to describe a distribution in brief terms there
exist additional measures, such as the variance which measures the dispersion
or spread, and moments.
For calculating the moments of uncertain random variable, some formulas are provided through chance distribution and inverse chance distribution. The main results are explained by using several examples.


[1] X. Chen and W. Dai, Maximum entropy principlefor uncertain variables, International Jour-
nal of Fuzzy Systems, 13(3) (2011), 232{236.
[2] X. Chen, S. Kar and D. Ralescu, Cross-entropy measure of uncertain variables, Information
Sciences, 201 (2012), 53{60.
[3] X. Chen and D. Ralescu, Liu process and uncertain calculus, Journal of Uncertainty Analysis
and Applications, 1(3) (2013), 1{ 12.
[4] W. Dai and X. Chen, Entropy of function of uncertain variables, Mathematics and Computer
Modelling, 55 (2012), 754{760.
[5] H. Y. Guo and X. S. Wang, Variance of uncertain random variables, Journal of Uncertainty
Analysis and Applications, 2(6) (2014), 1{7.
[6] Y. C. Hou, Subadditivity of chance measur, Journal of Uncertainty Analysis and Applications,
2(14) (2014), 1{8.
[7] A. N. Kolmogorov, Grundbegri e der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin,
[8] R. Kruse and K. Meyer, Statistics with Vague Data, Reidel Publishing Company, Dordrecht,
[9] B. Liu, Uncertainty Theory, 5th ed., 2014.
[10] B. Liu, Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007.
[11] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 31
(2009), 3{10.
[12] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty,
Springer-Verlag, Berlin, 2010.
[13] B. Liu, Toward uncertain nance theory, Journal of Uncertainty Analysis and Applications,
1(1) (2013), 1{15.
[14] Y. H. Liu, Uncertain random variables: A mixture of uncertainty and randomness, Soft
Computing, 17(4) (2013), 625{634.
[15] Y. H. Liu, Uncertain random programming with applications, Fuzzy Optimization and Deci-
sion Making, 12(2) (2013), 153{169.
[16] Y. H. Liu and M. H. Ha, Expected value of function of uncertain variables, Journal of Un-
certain Systems, 4(3) (2010), 181{186.
[17] Y. K. Liu and B. Liu, Fuzzy random variables: a scalar expected value operator, Fuzzy
Optimization and Decision Making, 2(2) (2003), 143{160.
[18] Y. K. Liu and B. Liu, Fuzzy random programming with equilibrium chance constraints, In-
formtion Sciences 170 (2005), 363{395.
[19] Z. X. Peng and K. Iwamura, A sucient and necessary condition of uncertainty distribution,
Journal of Interdisciplinary Mathematics, 13(3) (2010), 277{285.
[20] M. Puri and D. Ralescu, Fuzzy random variables, Journal of Mathmatical Application, 114
(1986), 409{422.
[21] Y. H. Sheng and S. Kar, Some results of moments of uncertain variable through inverse
uncertainty distribution, Fuzzy Optimization and Decision Making, 14 (2015), 57{76.
[22] Y. H. Sheng and K. Yao, Some formulas of variance of uncertain random variable, Journal
of Uncertainty Analysis and Applications, 2(12) (2014), 1{10.
[23] J. L. Teugels and B. Sundt, Encyclopedia of actuarial science, Wiley & Sons, 1 (2004).
[24] M. Wen and R. Kang, Reliability analysis in uncertain random system, Fuzzy Optimization
and Decision Making, doi:10.1007/s10700-016-9235-y, (2016).
[25] K. Yao, A formula to calculate the variance of uncertain variable, Soft Computing, 19(10)
(2015), 2947{2953.