Application of Probability Theory and Mathematical Statistics in Big Data Analysis

With the rapid growth of the Internet, the Internet of Things, and digital platforms, large-scale data analysis has become increasingly important in many practical fields. However, large datasets still contain randomness, variability, and uncertainty, which require appropriate statistical methods for reliable interpretation. This study examines the application of probability theory and mathematical statistics in big data analysis from the perspectives of sampling distributions, numerical estimation, and parameter inference. Three statistical simulation experiments were carried out using Python to examine the Central Limit Theorem, Monte‑Carlo estimation, and confidence‑interval analysis over a range of sample‑size conditions. The results show that increasing the sample size significantly improves statistical stability. The standard deviation of sample means decreased from 0.6337 to 0.0906, the mean absolute Monte Carlo estimation error declined from 0.12510 to 0.00394, and the average width of the 95% confidence interval decreased from 7.3884 to 0.5566. These findings indicate that probability theory and mathematical statistics remain fundamental tools for describing uncertainty, evaluating estimation accuracy, and supporting reliable analysis in large-scale data environments.