On some new properties of the beta distribution

✍ Scribed by Włodzimierz Krysicki

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 94 KB
• Volume: 42
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(98)00197-7

✦ Synopsis

The ÿrst aim of this paper is to show how to present a random variable with the beta distribution (of the ÿrst kind) as a ÿnite or inÿnite product of independent random variable's (r.v.)'s X k , where k ∈ [1; 2; : : : ; n] or k ∈ N. Such a presentation of an r.v. with the gamma distribution in the form of an inÿnite 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 deÿne the square of the Vandermonde determinant with random elements. Next, the convergence of the series ∞ 1 (1=2 k )ln X k to ln X with probability one has been proved. Finally, the Mieshalkin-Rogozin theorem [Mieshalkin, D., Rogozin, B.A., 1963. Ocenka razstajania mie żdu funkcjami raspredelenia po blizosti ich charakteristiczeskich funkcji i jeje primenenie k centralnoj predielnoj teoremie. (in Predelnyje teoremy verojatnostej. Taszkent. Akad Nauk Uzbec. SSR)], modiÿed in this paper, has been applied to the beta distribution.