A first–passage time random walk distribution with five transition probabilities: a generalization of the shifted inverse trinomial

In this paper a univariate discrete distribution, denoted by GIT, is proposed as a generalization of the shifted inverse trinomial distribution, and is formulated as a first-passage time distribution of amodified randomwalk on the half-plane with five transition probabilities. In contrast, the inverse trinomial arises as a random walk on the real line with three transition probabilities. The probability mass function (pmf) is expressible in terms of the Gauss hypergeometric function and this offers computational advantage due to its recurrence formula. The descending factorial moment is also obtained. The GIT contains twenty-two possible distributions in total. Special cases include the binomial, negative binomial, shifted negative binomial, shifted inverse binomial or, equivalently, lost-games, and shifted inverse trinomial distributions.A subclassGIT3,1 is a particular member of Kemp’s class of convolution of pseudo-binomial variables and its properties such as reproductivity, formulation, pmf, moments, index of dispersion, and approximations are studied in detail. Compound or generalized (stopped sum) distributions provide inflated models. The inflated GIT3,1 extends Minkova’s inflated-parameter binomial and negative binomial. Abivariatemodel which has theGIT as amarginal distribution is also proposed. K. Aoyama School of Fundamental Science and Technology, Keio University, Yokohama 223-8522, Japan e-mail: [email protected] K. Shimizu (B) Department of Mathematics, Keio University, Yokohama 223-8522, Japan e-mail: [email protected] S. H. Ong Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia e-mail: [email protected]