IJPAM: Volume 23, No. 1 (2005)

ON THE EQUALITY IN DISTRIBUTION OF
THE RANDOM VARIABLES $X$ AND $g(X)$

Dan Kucerovsky$^1$, Eric Marchand$^2$, Robert D. Small$^3$
$^{1,3}$Department of Mathematics and Statistics
University of New Brunswick
P.0. Box 4400, Fredericton, N.B., E3B 5A3, CANADA
$^1$e-mail: dan@math.unb.ca
$^3$e-mail: don@math.unb.ca
$^2$Department of Mathematics
Faculty of Science
University of Sherbrooke
2500 Boul. de l'Université
Sherbrooke, Québec, J1K 2R1, CANADA
e-mail: eric.marchand@usherbrooke.ca


Abstract.We study the equidistributional identity \begin{equation*}X=^dg(X) \end{equation*} in $g(X)$ for continuous random variables $X$ in the case of one-to-one and onto functions $g:\Re\longrightarrow\Re$ with at most one singularity. Such equidistributional identities and the related characterizations are not only of intrinsic interest, but also offer potential usefulness in the identification of probability models. We obtain a number of very general characterization results, one of which specializes to give a new characterization of the Cauchy distribution (Theorem [*]).

Received: July 13, 2005

AMS Subject Classification: 60E10, 60E05, 39B05

Key Words and Phrases: characterizations, equidistributional identities, self-inverses, functional equations

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 23
Issue: 1