Calibrated estimates of uncertainty are critical for many real-world computer vision applications of deep learning. While there are several widely-used uncertainty estimation methods, dropout inference stands out for its simplicity and efficacy. This technique, however, requires multiple forward passes through the network during inference and therefore can be too resource-intensive to be deployed in real-time applications. We propose a simple, easy-to-optimize distillation method for learning the conditional predictive distribution of a pre-trained dropout model for fast, sample-free uncertainty estimation in computer vision tasks. We empirically test the effectiveness of the proposed method on both semantic segmentation and depth estimation tasks and demonstrate our method can significantly reduce the inference time, enabling real-time uncertainty quantification, while achieving improved quality of both the uncertainty estimates and predictive performance over the regular dropout model.