In this paper we are interested in a class of fuzzy numbers which is uniquely identified by their membership functions. The function space, denoted Xh,p, will be constructed combining nonlinear mappings h (subjective perception) and probability density functions (PDF) p (objective entity), respectively. Under our assumptions, prove that there always exists to fulfill the observed outcome for gi...

Distributions have several different types of associated functions. The most familiar is the density function. Figure 1A and 1D show density functions for the normal distribution and gamma distribution, respectively. Density functions are related to histograms in that if enough data are collected then the histogram and density share the same shape. Hence, density functions serve as a first-orde...

Mixture of experts (MoE) models are widely applied for conditional probability density estimation problems. We demonstrate the richness class MoE by proving denseness results in Lebesgue spaces, when inputs and outputs variables both compactly supported. further prove an almost uniform convergence result input is univariate. Auxiliary lemmas proved regarding soft-max gating function class, thei...

The Baranyi–Roberts model describes the dynamics of volumetric densities two interacting cell populations. We randomize this by considering that initial conditions are random variables whose distributions determined using sample data and principle maximum entropy. Subsequenly, we obtain Liouville–Gibbs partial differential equation for probability density function two-dimensional solution stoch...