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System-level decision making in transportation needs to understand day-to-day variation of network flows, which calls for accurate modeling and estimation of probabilistic dynamic travel demand on networks. Most existing studies estimate deterministic dynamic origin-destination (OD) demand, while the day-to-day variation of demand and flow is overlooked. Estimating probabilistic distributions of dynamic OD demand is challenging due to the complexity of the spatio-temporal networks and the computational intensity of the high-dimensional problems. With the availability of massive traffic data and the emergence of advanced computational methods, this paper develops a data-driven framework that solves the probabilistic dynamic origin-destination demand estimation (PDODE) problem using multi-day data. Different statistical distances (e.g., lp-norm, Wasserstein distance, KL divergence, Bhattacharyya distance) are used and compared to measure the gap between the estimated and the observed traffic conditions, and it is found that 2-Wasserstein distance achieves a balanced accuracy in estimating both mean and standard deviation. The proposed framework is cast into the computational graph and a reparametrization trick is developed to estimate the mean and standard deviation of the probabilistic dynamic OD demand simultaneously. We demonstrate the effectiveness and efficiency of the proposed PDODE framework on both small and real-world networks. In particular, it is demonstrated that the proposed PDODE framework can mitigate the overfitting issues by considering the demand variation. Overall, the developed PDODE framework provides a practical tool for public agencies to understand the sources of demand stochasticity, evaluate day-to-day variation of network flow, and make reliable decisions for intelligent transportation systems.