This work studies probability functions that appear in stochastic programming models. Although their differentiability has been widely investigated, the Lipschitz continuity of their gradients, crucial for the design and analysis of modern optimization algorithms, has received little attention. We develop a general framework that ensures differentiability and gradient Lipschitz continuity under practical conditions. Our framework unifies and extends existing results and applies to a broad class of continuous distributions. Building on this theory, we propose a specialized method for a class of probability maximization problems. Our approach handles general distributions and enjoys convergence guarantees. Numerical experiments against state‑of‑the‑art methods demonstrate clear gains in accuracy, efficiency, and scalability.