Building robust, interpretable, and secure artificial intelligence system requires some degree of quantifying and representing uncertainty via a probabilistic perspective, as it allows to mimic human cognitive abilities. However, probabilistic computation presents significant challenges due to its inherent complexity. In this paper, we develop an efficient and interpretable probabilistic computation framework by truncating the probabilistic representation up to its first two moments, i.e., mean and covariance. We instantiate the framework by training a deterministic surrogate of a stochastic network that learns the complex probabilistic representation via combinations of simple activations, encapsulating the non-linearities coupling of the mean and covariance. We show that when the mean is supervised for optimizing the task objective, the unsupervised covariance spontaneously emerging from the non-linear coupling with the mean faithfully captures the uncertainty associated with model predictions. Our research highlights the inherent computability and simplicity of probabilistic computation, enabling its wider application in large-scale settings.