English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Let $T\$ be a stopping time associated with a sequence of independent randomvariables $Z_{1},Z_{2},...$ . By applying a suitable change in the probabilitymeasure we present relations between the moment or probability generatingfunctions of the stopping time $T$ and the stopped sum$%S_{T}=Z_{1}+Z_{2}+...+Z_{T}$. These relations imply that, when thedistribution of $S_{T}$\ is known, then the distribution of $T$\ is also knownand vice versa. Applications are offered in order to illustrate theapplicability of the main results, which also have independent interest. In thefirst one we consider a random walk with exponentially distributed up and downsteps 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 amanufacturing process and we let $Z_{i},i\geq 1$, denoting the number ofnon-conforming products in the $i$-th sample. We derive the joint distributionof the random vector $(T,S_{T})$, where $T$ is the waiting time until thesampling level of the inspection changes based on a $k$-run switching rule.Finally, we demonstrate how the joint distribution of $%(T,S_{T})$ can be usedfor the estimation of the probability $p$ of an item being defective, byemploying an EM algorithm.

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