Safe, socially compliant, and efficient navigation of low-speed autonomous vehicles (AVs) in pedestrian-rich environments necessitates considering pedestrians' future positions and interactions with the vehicle and others. Despite the inevitable uncertainties associated with pedestrians' predicted trajectories due to their unobserved states (e.g., intent), existing deep reinforcement learning (DRL) algorithms for crowd navigation often neglect these uncertainties when using predicted trajectories to guide policy learning. This omission limits the usability of predictions when diverging from ground truth. This work introduces an integrated prediction and planning approach that incorporates the uncertainties of predicted pedestrian states in the training of a model-free DRL algorithm. A novel reward function encourages the AV to respect pedestrians' personal space, decrease speed during close approaches, and minimize the collision probability with their predicted paths. Unlike previous DRL methods, our model, designed for AV operation in crowded spaces, is trained in a novel simulation environment that reflects realistic pedestrian behaviour in a shared space with vehicles. Results show a 40% decrease in collision rate and a 15% increase in minimum distance to pedestrians compared to the state of the art model that does not account for prediction uncertainty. Additionally, the approach outperforms model predictive control methods that incorporate the same prediction uncertainties in terms of both performance and computational time, while producing trajectories closer to human drivers in similar scenarios.