Distribution of Record Statistics in a q-Factorially Increasing Population

Abstract

The probability function and binomial moments of the number N n of (upper) records up to time (index) n in a q-factorially increasing population are obtained in terms of the non central signless q-Stirling numbers of the first kind. As a corollary, the probability function of the time T k of the kth record is also expressed in terms of the non central signless q-Stirling numbers of the first kind. The mean of T k is obtained as a q-series with terms of alternating sign. Finally, the probability function of the inter-record time W k = T k − T k−1 is obtained as a sum of a finite number of terms of q-numbers. The mean of W k is expressed by a q-series. As k increases to infinity the distribution of W k converges to a geometric distribution with failure probability q. Additional properties of the non central q-Stirling numbers of the first kind, which facilitate the present study, are derived.