This paper presents the exact mathematical derivation of the mean and variance properties for the Exponentially Weighted Moving Average (EWMA) statistic applied to binomial proportion monitoring in Multiple Stream Processes (MSPs). We develop a Cumulative Standardized Binomial EWMA (CSB-EWMA) formulation that provides adaptive control limits based on exact time-varying variance calculations, overcoming the limitations of asymptotic approximations during early-phase monitoring. The derivations are rigorously validated through Monte Carlo simulations, demonstrating remarkable agreement between theoretical predictions and empirical results. This work establishes a theoretical foundation for distribution-free monitoring of binary outcomes across parallel data streams, with applications in statistical process control across diverse domains including manufacturing, healthcare, and cybersecurity.

翻译：本文给出了在多流过程中应用于二项比例监控的指数加权移动平均统计量的均值与方差性质的精确数学推导。我们提出了一种累积标准化二项EWMA（CSB-EWMA）公式，该公式基于精确的时变方差计算提供自适应控制限，克服了早期阶段监控中渐近近似的局限性。推导过程通过蒙特卡洛模拟进行了严格验证，证明了理论预测与经验结果之间的显著一致性。这项工作为跨并行数据流的二元结果的无分布监控奠定了理论基础，可应用于包括制造业、医疗保健和网络安全在内的多个领域的统计过程控制。