Electronic system design based on deterministic techniques for power-temperature analysis is, in the context of current and future technologies, both unreliable and inefficient since the presence of uncertainty, in particular, due to process variation, is disregarded. In this paper, we propose a flexible probabilistic framework targeted at the quantification of the transient power and temperature variations of an electronic system. The framework is capable of modeling diverse probability laws of the underlying uncertain parameters and arbitrary dependencies of the system on such parameters. For the considered system, under a given workload, our technique delivers analytical representations of the corresponding stochastic power and temperature profiles. These representations allow for a computationally efficient estimation of the probability distributions and accompanying quantities of the power and temperature characteristics of the system. The approximation accuracy and computational time of our approach are assessed by a range of comparisons with Monte Carlo simulations, which confirm the efficiency of the proposed technique.