### 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)ealpha x, where u(x) is uniquely determined by the functions a(·) and b(·).

Key words and phrases:    Characterization, conditional expectation.

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