Discrete uniform and binomial distributions with infinite support
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METHODOLOGIES AND APPLICATION
Discrete uniform and binomial distributions with infinite support Andrey Pepelyshev1
· Anatoly Zhigljavsky1
© The Author(s) 2020
Abstract We study properties of two probability distributions defined on the infinite set {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 1 to all points in the set {0, 1, . . . , 1 − 1}, where 1 denotes the grossone. For we study is uniform and assigns masses 1/ d
this distribution, we study the problem of decomposing a random variable ξ with this distribution as a sum ξ = ξ1 + · · · + ξm , where ξ1 , . . . , ξm are independent non-degenerate random variables. Then, we develop an approximation for the probability 1 p) with p = c/ 1 α with 1/2 < α ≤ 1. The accuracy of this approximation mass function of the binomial distribution Bin(, is assessed using a numerical study. Keywords Binomial distribution · Poisson approximation · Charlier polynomials
1 Introduction In this paper, 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. 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 1 is a model of infinity which, as shown in denoted by , Sergeyev (2009), Sergeyev (2017) and many other publications can be very useful in solving diverse problems of computational mathematics and optimization; in such appli1 is used as numerical infinity. Grossone can also cations, be useful as a theoretical model of infinity, see, e.g., (Zhigljavsky 2012; Sergeyev 2017). Some historical, philosophical and logical aspects of grossone have been considered in Lolli (2012), Lolli (2015), Hansson (2020). In Sect. 1, we consider 1 and briefly discuss postulates of . For a positive integer n, the discrete uniform distribution on the set {0, 1, . . . , n − 1} assigns equal mass 1/n Communicated by V. Loia.
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Anatoly Zhigljavsky [email protected]
to all integers j ∈ {0, . . . , n − 1}. We use the notation DU(n) from Balakrishnan and Nevzorov (2004) for this distribution. An extension of this distribution to the infinite set {0, 1, 2 . . .} is denoted by DU(∞) and is used in Bayesian statistics as vague prior (often called ‘Jeffrey’s prior’) for large integer-valued parameters, in particular for the parameter N in the binomial distribution Bin (N , p), see, e.g., 1 is straight(Raftery 1988). An extension of DU(n) to DU() 1 we have forward: for a random variable (r.v.) ξ ∼DU(), 1 1 − 1}. This for all k ∈ {0, 1, . . . , Pr{ξ = k} = 1/() distribution has been considered, in particular, in Calude and 1 is easier than Dumitrescu (2020). The distribution DU() DU(∞): indeed, DU(∞) is a vague (improper) distribution 1 DU() 1 is but, if one agre
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