Maximum Statistics of N Random Variables Distributed by the Negative Binomial Distribution

Authors:
Peter J. Grabner;Helmut Prodinger
Affiliations:
Institut für Mathematik A, Steyrergasse 30, A–8010 Graz, Austria (e-mail: grabner@weyl.math.tu_graz.ac.at);Institut für Algebra und Diskrete Mathematik, Wiedner Hauptstrasse 8–10, A–1040 Wien, Austria (e-mail: Helmut.Prodinger@tuwien.ac.at)
Venue:
Combinatorics, Probability and Computing
Year:
1997

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Abstract

The lifetime of a player is defined to be the time where he gets his b-th hit, where a hit will occur with probability p. We consider the maximum statistics of N independent players. For b≠1 this is significantly more difficult than the known instance b=1. The expected value of the maximum lifetime of N players is given by logQ N+(b−1)logQ logQ N+ smaller order terms, where Q=1/(1−p).