In chip design, one of the main objectives is to decrease its clock cycle; however, the existing approaches to timing analysis under uncertainty are based on fundamentally restrictive assumptions. Statistical timing analysis techniques assume that the full probabilistic distribution of timing uncertainty is available; in reality, the complete probabilistic distribution information is often unavailable. Additionally, the existing alternative of treating uncertainty as interval-based, or affine, is limited since it cannot handle probabilistic information in principle. In this paper, a fundamentally new paradigm for timing uncertainty description is proposed as a way to consistently and rigorously handle partially available descriptions of timing uncertainty. The paradigm is based on a formal theory of interval probabilistic models that permit handling parameter uncertainty that is described in a distribution-free mode - just via the range, the mean, and the variance. This strategy permits effectively handling multiple real-life challenges, including imprecise and limited information about the distributions of process parameters, parameters coming from different populations, and the sources of uncertainty that are too difficult to handle via full probabilistic measures (e.g. on-chip supply voltage variation). Specifically, analytical techniques for bounding the distributions of probabilistic interval variables are proposed. Also, a provably correct strategy for fast Monte Carlo simulation based on probabilistic interval variables is introduced. A path-based timing algorithm implementing the novel modeling paradigm, as well as handling the traditional variability descriptions, has been developed. The results indicate the proposed technique can improve the upper bound of the 95th-percentile circuit delay, on average, by 4.8% across the ISCAS'85 benchmark circuits, compared to the worst-case timing analysis that uses only the interval information of the partially specified parameters.