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

TL;DR

This paper establishes relations between the distributions of stopping times and stopped sums for independent random variables, providing applications in random walks, manufacturing quality control, and defect probability estimation.

Contribution

It introduces a method using probability measure changes to relate the generating functions of stopping times and sums, enabling distribution determination of one from the other.

Findings

Derived the distribution of first exit times in a random walk with exponential steps.

Obtained the joint distribution of stopping time and sum in a quality control process.

Demonstrated estimation of defect probability using the joint distribution and EM algorithm.

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 $T$ and the stopped sum $%S_{T}=Z_{1}+Z_{2}+...+Z_{T}$. These relations imply that, when the distribution of $S_{T}$\ is known, 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 $Z_{i},i\geq 1$, 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 $T$ 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 $%(T,S_{T})$ can be used for the estimation of the probability $p$ of an item being defective, by employing an EM algorithm.