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• W2255068360 abstract "Let $T$ be a stopping time associated with a sequence of independent random variables $Z_{1},Z_{2},...$ . By applying a suitable change in the probability measure we present relations between the moment or probability generating functions of the stopping time and the stopped sum $%S_{T}=Z_{1}+Z_{2}+...+Z_{T}$. These relations imply that, when the distribution of $S_{T}$ 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 $Z_{i},igeq 1$, denoting the number of non-conforming products in the $i$-th sample. We derive the joint distribution of the random vector $(T,S_{T})$, where 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 $%(T,S_{T})$ can be used for the estimation of the probability $p$ of an item being defective, by employing an EM algorithm." @default.
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• W2255068360 date "2010-07-31" @default.
• W2255068360 modified "2023-09-27" @default.
• W2255068360 title "On the relation between the distributions of stopping time and stopped sum via Wald's Identity with applications" @default.
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