Monte Carlo simulations are widely used for uncertainty analysis in the probabilistic safety assessment of nuclear power plants. Despite many advantages, such as its general applicability, a Monte Carlo simulation has inherent limitations as a simulation-based approach. This study provides a mathematical formulation and analytic solutions for the uncertainty analysis in a probabilistic safety assessment (PSA). Starting from the definitions of variables, mathematical equations are derived for synthesizing probability density functions for logical AND, logical OR, and logical OR with rare event approximation of two independent events. The equations can be applied consecutively when there exist more than two events. For fail-to-run failures, the probability density function for the unavailability has the same probability distribution as the probability density function (PDF) for the failure rate under specified conditions. The effectiveness of the analytic solutions is demonstrated by applying them to an example system. The resultant probability density functions are in good agreement with the Monte Carlo simulation results, which are in fact approximations for those from the analytic solutions, with errors less than 12.6%. Important theoretical aspects are examined with the analytic solutions such as the validity of the use of a right-unbounded distribution to describe the uncertainty in the unavailability/probability. The analytic solutions for uncertainty analysis can serve as a basis for all other methods, providing deeper insights into uncertainty analyses in probabilistic safety assessment.