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

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).