Parametric clear sky models are often represented by simple analytic expressions that can efficiently generate plausible, natural radiance maps of the sky, taking into account expensive and hard to simulate atmospheric phenomena. In this work, we show how such models can be complemented by an equally simple, elegant and generic analytic continuous probability density function (PDF) that provides a very good approximation to the radiance-based distribution of the sky. We describe a fitting process that is used to properly parameterise a truncated Gaussian mixture model, which allows for exact, constant-time and minimal-memory sampling and evaluation of this PDF, without rejection sampling, an important property for practical applications in offline and real-time rendering. We present experiments in a standard importance sampling framework that showcase variance reduction approaching that of a more expensive inversion sampling method using Summed Area Tables.