###
A CHARACTERIZATION OF CERTAIN DISCRETE

EXPONENTIAL FAMILIES

###
T. T. NGUYEN, A. K. GUPTA AND Y. WANG

*Department of Mathematics and Statistics, Bowling Green State
University,*

Bowling Green, OH 43403-0221, U.S.A.
(Received February 22, 1995; revised August 2, 1995)

**Abstract.**
For independent random variables *X*
and *Y*, define *S* \equiv *X*+*Y*. When the conditional
expectations *E*[*g*(*X*) | *S*] \equiv *a*(*S*) and *E*[*h*(*X*) |
*S*] \equiv *b*(*S*) are given, then under certain assumptions, the
density function of *X* has the form of *u*(*x*)*k*(*alpha*)*e*^{alpha
x}, where *u*(*x*) is uniquely determined by the functions
*a*(·) and *b*(·).

*Key words and phrases*:
Characterization,
conditional expectation.

**Source**
( TeX ,
DVI ,
PS )