# The distributions of sum, minima and maxima of generalized geometric random variables

Ontology type: schema:ScholarlyArticle

### Article Info

DATE

2014-09-10

AUTHORS ABSTRACT

Geometric distribution of order k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} as one of the generalization of well known geometric distribution is the distribution of the number of trials until the first k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} consecutive successes in Bernoulli trials with success probability p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}. In this paper, it is shown that this generalized distribution can be represented as a discrete phase-type distribution. Using this representation along with closure properties of phase-type distributions, the distributions of sum, minima and maxima of two independent random variables having geometric distribution of order k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} are obtained. Numerical results are presented to illustrate the computational details. More... »

PAGES

1191-1203

### References to SciGraph publications

• 2009-08-08. On runs of length exceeding a threshold: normal approximation in STATISTICAL PAPERS
• 1996-12. On a waiting time distribution in a sequence of Bernoulli trials in ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS
• 2012-07-19. Exact distributions of constrained (k, ℓ) strings of failures between subsequent successes in STATISTICAL PAPERS
• 1984-12-01. On discrete distributions of orderk in ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS
• ### Journal

TITLE

Statistical Papers

ISSUE

4

VOLUME

56

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00362-014-0632-4

DOI

http://dx.doi.org/10.1007/s00362-014-0632-4

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1048519769

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