Uncertain statistics is a set of mathematical techniques for collecting, analyzing and interpreting data by uncertainty theory. There are mainly three modeling methods in uncertain statistics: uncertain time series analysis, uncertain regression analysis, and uncertain differential equation. This paper applies these tools to modeling China's per capita disposable income, and employs uncertain hypothesis test to determine whether the estimated uncertain statistical models fit China's per capita disposable income. In addition, this paper shows that it is necessary to use uncertain statistics instead of probability statistics to model China's per capita disposable income by investigating the corresponding residuals.

With the development of uncertain finance, uncertain stock models have become increasingly popular for modeling stock prices. This paper explores the symmetric properties inherent in the uncertain mean-reverting stock model, particularly in the structure of its differential equations and the resulting pricing formulas. The primary findings comprise the derivation of explicit pricing formulas, via uncertain differential equations, for European, American, Asian, and geometric average Asian options under the uncertain mean-reverting stock model. The symmetry in the inverse uncertainty distributions and the duality between call and put options are highlighted, demonstrating the model’s alignment with symmetric financial principles. Furthermore, several numerical examples are provided to illustrate the applicability and the symmetry-related characteristics of the derived formulas.

As a mathematical tool to rationally handle degrees of belief in human beings, uncertainty theory has been widely applied in the research and development of various domains, including science and engineering. As a fundamental part of uncertainty theory, uncertainty distribution is the key approach in the characterization of an uncertain variable. This paper shows a new formula to calculate the uncertainty distribution of strictly monotone function of uncertain variables, which breaks the habitual thinking that only the former formula can be used. In particular, the new formula is symmetrical to the former formula, which shows that when it is too intricate to deal with a problem using the former formula, the problem can be observed from another perspective by using the new formula. New ideas may be obtained from the combination of uncertainty theory and symmetry.