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 log Q N +( b −1)log Q log Q N + smaller order terms, where Q =1/(1− p ).