Probabilistic Modeling, Statistical Analysis, and Stochastic Optimization for Random Horizontal Ranges of Projectile

In this paper, we propose a simple and novel probabilistic model for random projectile paths with uniformly distributed launch angles. Specifically, we formulate a stochastic range maximization problem and derive a closed-form expression for probability density function (PDF) of the maximum horizontal range random variable. Using its PDF, we derive analytical results for various statistical parameters such as mean, standard deviation, median, and mode. We then present numerical results for those statistical parameters and interpret them qualitatively. Finally, we formulate a novel and insightful stochastic range optimization problem and present its solution. The proposed probabilistic model, its statistical analysis, and range optimization could be useful in modeling and analyzing events that involve motion like a projectile.