Discrete uniform and binomial distributions with infinite support

Abstract

AbstractWe study properties of two probability distributions defined on the infinite set$$\{0,1,2, \ldots \}$${0,1,2,…}and generalizing the ordinary discrete uniform and binomial distributions. Both extensions use the grossone-model of infinity. The first of the two distributions we study is uniform and assigns masses$$1/\textcircled {1}$$1/1to all points in the set$$ \{0,1,\ldots ,\textcircled {1}-1\}$${0,1,…,1-1}, where$$\textcircled {1}$$1denotes the grossone. For this distribution, we study the problem of decomposing a random variable$$\xi $$ξwith this distribution as a sum$$\xi {\mathop {=}\limits ^\mathrm{d}} \xi _1 + \cdots + \xi _m$$ξ=dξ1+⋯+ξm, where$$\xi _1 , \ldots , \xi _m$$ξ1,…,ξmare independent non-degenerate random variables. Then, we develop an approximation for the probability mass function of the binomial distribution Bin$$(\textcircled {1},p)$$(1,p)with$$p=c/\textcircled {1}^{\alpha }$$p=c/1αwith$$1/2<\alpha \le 1$$1/2<α≤1. The accuracy of this approximation is assessed using a numerical study.