The distributions of sum, minima and maxima of generalized geometric random variables

Abstract

Geometric distribution of order $$k$$ as one of the generalization of well known geometric distribution is the distribution of the number of trials until the first $$k$$ consecutive successes in Bernoulli trials with success probability $$p$$ . In this paper, it is shown that this generalized distribution can be represented as a discrete phase-type distribution. Using this representation along with closure properties of phase-type distributions, the distributions of sum, minima and maxima of two independent random variables having geometric distribution of order $$k$$ are obtained. Numerical results are presented to illustrate the computational details.