This study aims to optimize probabilistic inventory management by employing two distributions from the Topp–Leone family, namely the Topp–Leone–Shanker (TL–Sh) and the Topp–Leone–Rayleigh (TL–RR) distributions. The research focuses on estimating key indicators of the probabilistic Economic Order Quantity (EOQ) model, including the optimal order quantity, reorder point, and various inventory-related costs such as holding, shortage, ordering, and total expected costs. The empirical application is based on real production data from the Diyala State Company, represented by demand quantities for electrical transformers (11/250 KV) over the period 2019–2021. The parameters of the proposed distributions were estimated using the Maximum Product of Spacings (MPS) method. Goodness-of-fit criteria (AIC, AICc, and BIC) and statistical tests (Anderson–Darling, Cramér–von Mises, and Kolmogorov–Smirnov) confirmed that the TL–Sh distribution provides a better fit to the observed demand data compared to the TL–RR and baseline Shanker distributions. Based on these results, inventory model indicators were computed for both distributions. The findings reveal that the probabilistic inventory model based on the TL–Sh distribution yields significantly lower total expected costs compared to the TL–RR model, indicating its superior performance in capturing demand uncertainty and optimizing inventory decisions. The study highlights the importance of selecting appropriate probability distributions in stochastic inventory models and demonstrates the effectiveness of the TL–Sh distribution in improving inventory control and reducing overall costs.