Beamforming is a fundamental technology that not only enhances communication efficiency but also lays the foundation for massive multiple-input multiple-output~(MIMO) systems. However, its reliance on accurate channel state information (CSI) estimation introduces significant training overhead and feedback costs, especially in large-scale antenna systems. In this paper, we investigate positioning-assisted beamforming as a competitive alternative to the CSI-based methods, which circumvents the complicated CSI estimation. In particular, we analyze the outage probability of positioning-assisted systems with joint Gaussian beams and derive its closed-form expressions for both two-dimensional~(2D) and three-dimensional~(3D) scenarios. Based on these results, we also derive closed-form expressions for the optimal joint Gaussian beam pattern. The optimal solution is independent of the positioning error distribution in 2D scenarios but depends on it in 3D cases. Subsequently, the asymptotic performance of the approximation error is analyzed. Numerical results verify the derived outage probability expressions, and show the effectiveness of the beam pattern optimization.