On the relation between the distributions of stopping time and stopped sum via Wald's Identity with applications

Let $T$ be a stopping time associated with a sequence of independent random variables $Z_{1},Z_{2},...$ . By employing a version of Wald's likelihood ratio identity we present relations between the moment or probability generating functions of the stopping time $T$ and the stopped sum $S_{T}=Z_{1}+Z_{2}+...+Z_{T}$. These relations imply that, when the distribution of $S_{T}$ is known (for all probability measures derived after exponentially tilting the original probability measure), then the distribution of $T$ is also known and vice versa. Two 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 $$, 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 $T$ is the waiting time until the sampling level of the inspection changes based on a $k$-run switching rule.