The clearness index for hourly and daily radiation is an important parameter in describing solar radiation. Liu and Jordan demonstrated that the monthly average daily clearness index could be used to predict the long-term distribution of daily solar radiation in a month. This paper reviews recent literature on the prediction of hourly and daily frequency distributions and cumulative frequency distributions of clearness indices. Ten years of measured weather data for six cities in the US are used to investigate the nature of the hourly and daily frequency distributions. A second set of ten years of data for six cities is used to verify the predictions. A bi-exponential probability density function is proposed that fits the observed bimodal nature of the data better than existing models. A case is made for the function being universal.