Reliability analysis of engineering systems has traditionally been done using computationally expensive computer simulations that cannot attain 100% accuracy due to their inherent limitations. The authors conduct a formal reliability analysis using higher-order-logic theorem proving, which is known to be sound, accurate, and exhaustive. For this purpose, they present the higher-order-logic formalization of independent multiple continuous random variables, their verified probabilistic properties, and generalized relations for commonly encountered reliability structures in engineering systems. To illustrate the usefulness of the approach, the authors present the formal reliability analysis of a single stage transmission of an automobile.