This paper presents a hierarchical approach for estimating the mission feasibility, i.e., the probability of mission completion (PoMC), for mobile robotic systems operating in stochastic environments. Mobile robotic systems rely on onboard energy sources that are expended due to stochastic interactions with the environment. Resultantly, a bivariate distribution comprised of energy source (e.g., battery) run-time and mission time marginal distributions can be shown to represent a mission process that characterizes the distribution of all possible missions. Existing methodologies make independent stochastic predictions for battery run-time and mission time. The approach presented makes use of the marginal predictions, as prediction pairs, to allow for Bayesian correlation estimation and improved process characterization. To demonstrate both prediction accuracy and mission classification gains, the proposed methodology is validated using a novel experimental testbed that enables repeated battery discharge studies to be conducted as a small robotic ground vehicle traverses stochastic laboratory terrains.