Probability, Waiting-Time Results for Pattern and Frequency Quotas in the Same Inverse Sampling Problem via the Dirichlet

Abstract

Several papers have been written recently on the study of probabilities and waiting-time questions associated with inverse sampling schemes in a multinomial setting. The stopping rule used has been either reaching some frequency (or quota) in a particular cell and/or reaching some run of length r in some other cell disjoint from the former cell. In some cases we considered frequency quotas for some cells and run quotas for other cells in the same problem and stopped (for example) with the soonest of these events. In the present paper we replace the run quota by its natural (but by no means trivial) generalization: the attainment of some preassigned pattern. The pattern problem has been studied by different authors, most notably by Feller [2], but none of these authors have combined different patterns and frequency quotas in the same problem or have given explicit formulas for either case. Hence our general results can be checked against published results for special cases that are already in the literature. In this paper, we allow several different pattern quotas and/or sev­ eral frequency quotas in the same problem, provided they each involve disjoint cells; one (so-called) slack cell without any quota is also included. Explicit, simple, exact formulas are given in most cases. The stopping rule of principal interest is to wait for the soonest of all quotas; the Dirichlet integral is the principal tool used in the solution. Our formulation also allows several quota-free cells to be present and our results include the derivation of the probability generating function (pgf) and joint moments for the frequencies of quota-free cells at stopping time. It is interesting to note that in the Case 3 (described below) the correlation offrequencies is positive for all pairs of cell frequencies at stopping time, unlike the usual multinomial structure (with fixed sample size) where the answer is always negative.