This study introduces a generalized analytical framework for deriving the probability density function associated with observing a system’s state \( x \) over time \( t \) , based on a fractional diffusion-type differential equation defined on an unbounded domain. Employing methods from operational calculus and complex analysis-specifically, the residue theorem, we obtain an explicit closed-form expression for the amplitude distribution, capturing the temporal dynamics of time series over variable-length intervals. In contrast to conventional approaches, which typically restrict the fractional time derivative \( \beta \) to the range \( 0 < \beta < 1 \) and the spatial derivative \( \alpha \) to \( 1 < \alpha < 2 \) , the proposed model accommodates arbitrary positive orders of differentiation. This generalization substantially enhances the versatility of fractional models in time series analysis. The derived solution remains valid under the condition \( 0 < \beta /\alpha \le 0.865 \) , thereby extending its applicability to a broader class of nonlocal, memory-dependent dynamical systems. Empirical validation using electoral time series data from the 2008, 2012, and 2016 U.S. presidential campaigns confirms the model’s effectiveness and descriptive power.