Derivations for the Cumulative Standardized Binomial EWMA (CSB-EWMA) Control Chart

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.