The Elo rating system, originally developed by Arpad Elo, is a widely used method for assessing the skill levels of players in two-player games. This research focuses on analyzing the stationary distribution of player ratings within the Elo model. We developed a numerical algorithm to approximate the cumulative distribution function (CDF) of the stationary distribution and conducted extensive simulations to validate the accuracy and applicability of the algorithm. Our study included a grid convergence analysis, symmetry tests, and an examination of the "Elo conjecture", which posits a relationship between rating differences and match outcomes. The results confirmed the accuracy of the numerical scheme and revealed that the stationary distribution exhibits key properties, including symmetry around its mean and a dependence of variance on the parameter K, which controls the adjustment speed of ratings. These findings provide valuable insights into the Elo rating system's underlying dynamics, offering a foundation for both theoretical advancements and practical applications.