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The Maximum Negative Hypergeometric Distribution
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DOI：10.1007/s42519-025-00502-x.png)
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Abstract
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An urn contains a known number of balls of two different colors. We describe the random variable counting the smallest number of draws needed in order to observe at least c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,c\,$$\end{document} of both colors when sampling without replacement for a prespecified, positive integer value of c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,c$$\end{document}. This distribution is the finite sample analogy to the maximum negative binomial distribution described by Zhang et al. [14]. We describe the modes, approximating distributions, and estimation of the contents of the urn. This distribution is used to estimate the sample size for planning a stratified clinical trial.
Keywords:
Discrete distributions
Negative binomial distribution
Riff-shuffle distribution
Hypergeometric distribution
Negative hypergeometric distribution
Maximum negative binomial distribution
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