Moments of a class of discrete q-distributions

A wide class of q-distributions is defined on the stochastic model of a sequence of Bernoulli trials in which the conditional probability of success at the nth trial, given that k successes occur before that trial, varies geometrically with n and k. Let Xn be the number of successes up to nth trial and Yk be the number of trials until the occurrence of the kth success. The q-factorial moments of the random variables Xn, and Yk are derived. Further, the usual factorial moments of Xn, which are connected with the q-factorial moments through the q-Stirling numbers of the first kind, are deduced. Finally, the ascending factorial moments of Yk, in the particular case in which the success probability does not depend on n, are obtained through the probability generating function.