Spatial and temporal heterogeneity of ambient natural environments play a significant role in large-scale transport phenomena. Uncertainty about spatio-temporal fluctuations in system parameters (e.g., flow velocity) makes deterministic predictions of macroscopic system states (e.g., solute concentration) elusive. Distributions of system states generally exhibit highly non-Gaussian behavior, which cannot be captured solely by the corresponding mean and variance. Instead, these features of transport are described by the probability density function (PDF) of a system state, e.g., the PDF of concentration at a certain point in space and time. We study the PDF of a passive scalar that disperses in a random velocity field. We derive an explicit map between the velocity distribution and the scalar PDF, and obtain approximate solutions for the PDF of the normalized scalar. These solutions enable one to quantify explicitly the impact of dispersion on the evolution of the passive scalar PDF without recurrence to classical closure approximations used in mixing models.