Comprehensive Probability Distribution Fitting Toolkit with Maximum Likelihood Estimation Algorithms

This paper presents a comprehensive algorithm toolkit for fitting multiple probability distributions, featuring Maximum Likelihood Estimation (MLE) for optimal parameter determination, Least Squares Estimation (LSE) for curve fitting applications, and Expectation-Maximization (EM) algorithm implementations for Gaussian mixture model estimation. The EM algorithm implementation includes iterative parameter updates using expectation and maximization steps with convergence testing. Additionally, we provide practical test cases for EM algorithm validation and dedicated plotting functions that generate visual representations of each distribution fit, enabling intuitive understanding of algorithm performance through graphical output analysis. This toolkit finds wide applications in finance, statistical modeling, and machine learning domains, particularly for probability density estimation and parametric model fitting. Notably, our experimental results reveal interesting phenomena regarding convergence behavior and parameter sensitivity, which provide significant insights for further algorithm refinement and research development. We believe this work offers valuable references and inspiration for professionals and researchers in related fields, with practical code examples demonstrating distribution fitting techniques and parameter estimation methods.