On the relation between the distributions of stopping time and stopped sum with applications

M. V. Boutsikas; A. C. Rakitzis; D. L. Antzoulakos

paper

On the relation between the distributions of stopping time and stopped sum with applications

arXiv:1008.0116

Abstract

Let $T $ b e a s t o pp in g t im e a ssoc ia t e d w i t ha se q u e n ceo f in d e p e n d e n t r an d o m v a r iab l es$ Z_{1},Z_{2},... $. B y a ppl y in g a s u i t ab l ec han g e in t h e p r o babi l i t y m e a s u r e w e p r ese n t r e l a t i o n s b e tw ee n t h e m o m e n t or p r o babi l i t y g e n er a t in g f u n c t i o n so f t h es t o pp in g t im e$ T $an d t h es t o pp e d s u m$ %S_{T}=Z_{1}+Z_{2}+...+Z_{T} $. T h eser e l a t i o n s im pl y t ha t, w h e n t h e d i s t r ib u t i o n o f$ S_{T} $i s k n o w n, t h e n t h e d i s t r ib u t i o n o f$ T $i s a l so k n o w nan d v i ce v er s a . A ppl i c a t i o n s a r eo f f er e d in or d er t o i l l u s t r a t e t h e a ppl i c abi l i t y o f t h e main r es u l t s, w hi c ha l so ha v e in d e p e n d e n t in t er es t . I n t h e f i r s t o n e w eco n s i d er a r an d o m w a l k w i t h e x p o n e n t ia l l y d i s t r ib u t e d u p an dd o w n s t e p s an dd er i v e t h e d i s t r ib u t i o n o f i t s f i r s t e x i tt im e f r o manin t er v a l$ (-a,b). $I n t h eseco n d a ppl i c a t i o n w eco n s i d er a ser i eso f s am pl es f r o maman u f a c t u r in g p r ocess an d w e l e t$ Z_{i},i\geq 1 $, d e n o t in g t h e n u mb er o f n o n - co n f or min g p r o d u c t s in t h e$ i $- t h s am pl e . W e d er i v e t h e j o in t d i s t r ib u t i o n o f t h er an d o m v ec t or$ (T,S_{T}) $, w h er e$ T $i s t h e w ai t in g t im e u n t i l t h es am pl in g l e v e l o f t h e in s p ec t i o n c han g es ba se d o na$ k $- r u n s w i t c hin g r u l e . F ina l l y, w e d e m o n s t r a t e h o w t h e j o in t d i s t r ib u t i o n o f$ %(T,S_{T}) $c anb e u se df or t h ees t ima t i o n o f t h e p r o babi l i t y$ p$ of an item being defective, by employing an EM algorithm.

18 pages, 4 figures