Sums, differences, products, and ratios of hypergeometric beta variables

Nadarajah, Saralees

doi:10.1016/j.crma.2005.06.013

Statistics/Probability Theory

Sums, differences, products, and ratios of hypergeometric beta variables
[Sommes, différences, produits et rapports de variables hypergéométriques beta]

Présenté par : Paul Deheuvels

Nadarajah, Saralees ¹

¹ Department of Statistics, University of Nebraska, Lincoln, NE 68583, USA

Comptes Rendus. Mathématique, Tome 341 (2005) no. 2, pp. 129-132.

Résumé
Abstract

Dans des publications récentes, le Professeur T. Pham-Gia a calculé les distributions de sommes, différences, produits et rapports de variables aléatoires beta indépendantes. Dans cette Note, nous étendons les résultats du Professeur Pham-Gia au cas où $X_{1}$ and $X_{2}$ sont des variables aléatoires indépendantes obéissant aux distributions hypergéométriques confluente et de Gauss (qui sont des généralisations de la distribution beta). Pour chacune de ces distributions, nous dérivons des expressions excates des densités $S = X_{1} + X_{2}$ , $D = X_{1} - X_{2}$ , $P = X_{1} X_{2}$ et $R = X_{2} / X_{1}$ . Ces expressions impliquent des fonctions hypergéométriques de une et deux variables.

Recent papers by Professor T. Pham-Gia derived distributions of sums, differences, products and ratios of independent beta random variables. In this Note we extend Professor Pham-Gia's results when $X_{1}$ and $X_{2}$ are independent random variables distributed according to the confluent and Gauss hypergeometric distributions (which are generalizations of the beta distribution). For each of these distributions, we derive exact expressions for the densities of $S = X_{1} + X_{2}$ , $D = X_{1} - X_{2}$ , $P = X_{1} X_{2}$ , and $R = X_{2} / X_{1}$ . The expressions turn out to involve the hypergeometric functions of one and two variables.

Reçu le : 2004-10-28
Accepté le : 2005-06-02
Publié le : 2005-07-15

DOI : 10.1016/j.crma.2005.06.013

Affiliations des auteurs :

Nadarajah, Saralees ¹

¹ Department of Statistics, University of Nebraska, Lincoln, NE 68583, USA

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Nadarajah, Saralees. Sums, differences, products, and ratios of hypergeometric beta variables. Comptes Rendus. Mathématique, Tome 341 (2005) no. 2, pp. 129-132. doi : 10.1016/j.crma.2005.06.013. http://archive.numdam.org/articles/10.1016/j.crma.2005.06.013/

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