The assumption that probability density functions of daily clearness indices are unimodal was tested. Bimodal behaviour was observed in almost 60% of 600 monthly data sets and was shown to be the most usual shape at 50 locations between latitudes 18.43degreesN and 64.81degreesN and at between 2 and 2297 m above sea level. A bi-exponential probability density function was proposed that fits the observed behaviour of daily clearness indices. The proposed function uses the mean monthly clearness index (K) over bar (t), and the mean monthly solar height at noon (h) over bar (max) as inputs. The bi-exponential method for predicting daily distributions is shown to reduce the RMS error by over 20% compared with earlier methods. Parametric representations of monthly maximum and minimum clearness index using (K) over bar (t) and (h) over bar (max) are combined with the bi-exponential function to obtain a bi-variable probability density function f(K-t, (K) over bar (t), (h) over bar (max)) and a cumulative distribution function F(K-t, (K) over bar (t), (h) over bar (max)), analogous to those of Liu and Jordan. (C) 2003 Elsevier Ltd. All rights reserved.