2010/07/31 by M. V. Boutsikas, Boutsikas, M. V., A. C. Rakitzis +3
Mathematics · #62E15 #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #msc:62E15 #primary:60G40 #secondary:60G50 #stat.TH
paper · pdf · doi:10.48550/arxiv.1008.0116
18 pages, 4 figures
arxiv created 2011/06/27 · arxiv updated 2011/06/28
Let T$ be a stopping time associated with a sequence of independent random variables Z1,Z2,... . By applying a suitable change in the probability measure we present relations between the moment or probability generating functions of the stopping time T and the stopped sum %ST=Z1+Z2+...+ZT. These relations imply that, when the distribution of ST is known, then the distribution of T is also known and vice versa. Applications are offered in order to illustrate the applicability of the main results, which also have independent interest. In the first one we consider a random walk with exponentially distributed up and down steps and derive the distribution of its first exit time from an interval (-a,b). In the second application we consider a series of samples from a manufacturing process and we let Zi,i≥ 1, denoting the number of non-conforming products in the i-th sample. We derive the joint distribution of the random vector (T,ST), where T is the waiting time until the sampling level of the inspection changes based on a k-run switching rule. Finally, we demonstrate how the joint distribution of %(T,ST) can be used for the estimation of the probability p of an item being defective, by employing an EM algorithm.