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On some new properties of the beta distribution

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  • Krysicki, Wlodzimierz

Abstract

The first aim of this paper is to show how to present a random variable with the beta distribution (of the first kind) as a finite or infinite product of independent random variable's (r.v.)'s Xk, where k[set membership, variant][1,2,...,n] or . Such a presentation of an r.v. with the gamma distribution in the form of an infinite product of r.v.'s was used by Lu and Richards [Lu, I-Li, Richards, D., 1993. Random discriminants. Ann. Statist. 21 (4) 1992-2000] to define the square of the Vandermonde determinant with random elements. Next, the convergence of the series [summation operator]1[infinity](1/2k)ln Xk to ln X with probability one has been proved. Finally, the Mieshalkin-Rogozin theorem [Mieshalkin, D., Rogozin, B.A., 1963. Ocenka razstajania miedu funkcjami raspredelenia po blizosti ich charakteristiczeskich funkcji i jeje primenenie k centralnoj predielnoj teoremie. (in Predelnyje teoremy verojatnostej. Taszkent. Akad Nauk Uzbec. SSR)], modified in this paper, has been applied to the beta distribution.

Suggested Citation

  • Krysicki, Wlodzimierz, 1999. "On some new properties of the beta distribution," Statistics & Probability Letters, Elsevier, vol. 42(2), pages 131-137, April.
    • Handle: RePEc:eee:stapro:v:42:y:1999:i:2:p:131-137
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    Cited by:

    1. Jones, M.C. & Balakrishnan, N., 2021. "Simple functions of independent beta random variables that follow beta distributions," Statistics & Probability Letters, Elsevier, vol. 170(C).
    2. Yury R. Benites & Vicente G. Cancho & Edwin M. M. Ortega & Roberto Vila & Gauss M. Cordeiro, 2022. "A New Regression Model on the Unit Interval: Properties, Estimation, and Application," Mathematics, MDPI, vol. 10(17), pages 1-17, September.

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