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A Discrete Probability Distribution and Some Applications

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Abstract

This paper is concerned with the probability distribution \(a_{n,j}:=4^{-n}{2j\atopwithdelims ()j}{2n-2j\atopwithdelims ()n-j}, j=0,1,\dots ,n\). We present basic properties of the sequence \((a_{n,j})\): integral representations, recurrence formulas, convexity properties, bounds for the associated information potential. Two related random variables are also studied and compared from the point of view of the stochastic convex ordering and in connection with strongly convex functions. We consider a quadrature formula and investigate it with analytic and probabilistic methods. The paper contains two open problems.

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Acknowledgements

This work was supported by a Hasso Plattner Excellence Research Grant (LBUS-HPI-ERG-2020-07), financed by the Knowledge Transfer Center of the Lucian Blaga University of Sibiu.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Str. Dr. I. Ratiu, No.5-7, 550012, Sibiu, Romania

    Ana-Maria Acu

  2. Department of Mathematics, Faculty of Automation and Computer Science, Technical University of Cluj-Napoca, Str. Memorandumului nr. 28, 400114, Cluj-Napoca, Romania

    Ioan Rasa

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  1. Ana-Maria Acu
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Correspondence to Ioan Rasa.

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Note added in proof.

The authors remarked that inequality (10.3) can be proved using Theorem 10.1. Details will appear in a forthcoming paper.

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Acu, AM., Rasa, I. A Discrete Probability Distribution and Some Applications. Mediterr. J. Math. 20, 34 (2023). https://doi.org/10.1007/s00009-022-02243-8

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  • DOI: https://doi.org/10.1007/s00009-022-02243-8

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