Several Inequalities on Moment of Uncertain Variables

摘要

An uncertain variable is a Borel measurable function whose domain is uncertainty space and range is the set of real numbers. However, for many reasons, like the difficult of collecting data, the value of an uncertain variable is usually not easy to measure accurately. Hence many scholars study the estimation range of the value of an uncertain variable, and usually to estimate the upper or lower bounds of the moment of an uncertain variable is the primary idea. Many inequalities are established to estimate the above bounds, but there are still some problems on the estimation of the moment of uncertain variables. For instance, the even-order moment of an uncertain variable cannot be uniquely calculated at present. So the aim of this paper is to estimate the upper or power bounds of the moment of uncertain variables or the uncertain measure of an event by establishing several new inequalities. Firstly, we extend the Lyapunov inequality on uncertain variable and this inequality gives the upper bound of the even-order moment of an uncertain variable, and as a corollary, the lower bound of the above even-order moment is given. Then the inequality of arithmetic-geometry is proved, which estimates the lower bound of the expected value of an uncertain variable. After that, two equivalent inequalities are given, which can be used to judge the existence of the expected value of a function of an uncertain variable. Finally, as for two independent and identically distributed uncertain variables, the weakly symmetric inequalities are investigated to estimate the upper and lower bounds of the uncertainty distributions of the difference of these uncertain variables which implies the uncertain measures of several events. The above inequalities extend the application range of uncertain variable.