{
  "id": "1008.0116",
  "version": "v2",
  "published": "2010-07-31T18:02:20.000Z",
  "updated": "2011-06-27T09:50:29.000Z",
  "title": "On the relation between the distributions of stopping time and stopped sum with applications",
  "authors": [
    "M. V. Boutsikas",
    "A. C. Rakitzis",
    "D. L. Antzoulakos"
  ],
  "comment": "18 pages, 4 figures",
  "categories": [
    "math.ST",
    "stat.TH"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-27T09:50:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G40",
      "60G50",
      "62E15"
    ],
    "keywords": [
      "stopping time",
      "stopped sum",
      "application",
      "joint distribution",
      "independent random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.0116B"
    }
  }
}