New statistical distributions for group counting in Bernoulli and Poisson processes

Based on a sequential group sampling procedure, the authors introduce a new statistical distribution for group-by-group counting for Bernoulli processes. Suppose that a population contains an infinite number of items, each item having probability p for nonconformity and probability 1-p for conformity. Items are inspected group by group, where each group contains m items, and the number of nonconforming items are recorded only when inspection of a group is completed. The inspection procedure is terminated if the cumulative number of nonconforming items recorded is greater than or equal to a specified number r. The distribution introduced in this paper consists of two random variables: the number of groups inspected, K, and the cumulative number of nonconforming items observed, X. Statistical properties and inferences for the bivariate distribution of (K,X) are studied. The authors also discuss the limiting case as m → ∞ and p → 0 with mp fixed, and derive a new bivariate distribution for group-by-group counting for Poisson processes that approaches the gamma distribution in limit. A new type of control chart and an acceptance sampling procedure are developed based on (K,X), and examples are given to illustrate their applications.