Discrete uniform and binomial distributions with infinite support

We study properties of two probability distributions defined on the infinite set $$\{0,1,2, \ldots \}$$ 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 $$1/\textcircled {1}$$ to all points in the set $$ \{0,1,\ldots ,\textcircled {1}-1\}$$ , where $$\textcircled {1}$$ denotes the grossone. For this distribution, we study the problem of decomposing a random variable $$\xi $$ with this distribution as a sum $$\xi {\mathop {=}\limits ^\mathrm{d}} \xi _1 + \cdots + \xi _m$$ , where $$\xi _1 , \ldots , \xi _m$$ are independent non-degenerate random variables. Then, we develop an approximation for the probability mass function of the binomial distribution Bin $$(\textcircled {1},p)$$ with $$p=c/\textcircled {1}^{\alpha }$$ with $$1/2<\alpha \le 1$$ . The accuracy of this approximation is assessed using a numerical study.

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