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Approximation and uncertainty quantification methods based on Lagrange interpolation are typically abandoned in cases where the probability distributions of one or more system parameters are not normal, uniform, or closely related distributions, due to the computational issues that arise when one wishes to define interpolation nodes for general distributions. This paper examines the use of the recently introduced weighted Leja nodes for that purpose. Weighted Leja interpolation rules are presented, along with a dimension-adaptive sparse interpolation algorithm, to be employed in the case of high-dimensional input uncertainty. The performance and reliability of the suggested approach is verified by four numerical experiments, where the respective models feature extreme value and truncated normal parameter distributions. Furthermore, the suggested approach is compared with a well-established polynomial chaos method and found to be either comparable or superior in terms of approximation and statistics estimation accuracy