In many scientific areas, a deterministic model (e.g., a differential equation) is equipped with parameters. In practice, these parameters might be uncertain or noisy, and so an honest model should provide a statistical description of the quantity of interest. Underlying this computational question is a fundamental one - If two ``similar" functions push-forward the same measure, are the new resulting measures close, and if so, in what sense? I will first show how the probability density function (PDF) can be approximated, using spectral and local methods, and present applications to nonlinear optics. We will then discuss the limitations of PDF approximation, and present an alternative Wasserstein-distance formulation of this problem, which yields a much simpler theory.