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This paper aims to present a study about some properties of the fractional probability measure. We define the fractional moments generating function and deduce its properties. We provide various results and show their application in an illustrated example. Moreover, this study confirms that the theorem of continuity of probability measure can not be achieved for the fractional probability measure.

[1] This paper presents the results of statistical investigations for the evaluation of soil water retention models (SWRMs). We employed three different methods developed for model selection in the field of nonlinear regression, namely, simulation studies, analysis of nonlinearity measures, and resampling strategies such as cross validation and bootstrap methods. Using these methods together with small data sets, we evaluated the performance of three exemplarily chosen types of SWRMs with respect to their parameter properties and the reliability of model predictions. The resulting rankings of models show that the favorable models are characterized by few parameters with an almost linear estimation behavior and close to symmetric distributions. To further demonstrate the potential of the statistical methods in the field of model selection, a modification of the four-parameter van Genuchten model is proposed which shows significantly improved and robust statistical properties.