Abstract
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.
Highlights
We are interested in properties of two probability distributions defined on the infinite set {0, 1, 2, . . .} and generalizing the ordinary discrete uniform and binomial distributions
We study properties of two probability distributions defined on the infinite set {0, 1, 2, . . .} and generalizing the ordinary discrete uniform and binomial distributions
We study the accuracy of different approximations for the probability mass function of the binomial distribution Bin( 1, p) with p = c/ 1 α with 1/2 < α ≤ 1
Summary
We are interested in properties of two probability distributions defined on the infinite set {0, 1, 2, . . .} and generalizing the ordinary discrete uniform and binomial distributions. . .} and generalizing the ordinary discrete uniform and binomial distributions. Both of these extensions have been recently discussed in Calude and Dumitrescu (2020) and mentioned in Zhigljavsky (2012); both extensions use the notion of grossone. The grossone, introduced in Sergeyev (2013) and denoted by 1 , is a model of infinity which, as shown in Sergeyev (2009), Sergeyev (2017) and many other publications can be very useful in solving diverse problems of computational mathematics and optimization; in such applications, 1 is used as numerical infinity. Grossone can be useful as a theoretical model of infinity, see, e.g., (Zhigljavsky 2012; Sergeyev 2017). For a positive integer n, the discrete uniform distribution on the set {0, 1, .
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call