Explore a comprehensive lecture on statistical learning and probability distribution analysis. Delve into two major research directions concerning guaranteed generalizations from finite samples. Examine the challenge of learning under common prior knowledge assumptions, focusing on the sample complexity of learning Gaussian mixtures. Investigate the characterization of learnable distribution families, contrasting it with binary classification prediction and other machine learning tasks. Discover why learnability of distribution families cannot be characterized by a combinatorial dimension. Gain insights from collaborative research efforts, including work on Gaussian mixtures and distribution family learnability.