J. Ital. Statist. Soc. (1997) I. pp. 59-65 CHARACTERIZATIONS OF SOME CONTINUOUS DISTRIBUTIONS A. K. Gupta*, T. T. Nguyen and Y. Wang Bowling Green State University Abstract This paper gives characterizations of normal and gamma distributions via conditional structure. 1. Introduction The normal distribution is the most important distribution in statistics. Problems of conditional specifications ofbivariate normal distribution have attracted more and more attention in recent years. Suppose that both XIY and YlX have normal distribution then under some given condition, Brucker (1979), Fraser and Streit (1980), Castillo and Galambos (1989) characterized a bivariate normal distribu- tion. More general results are given by Cacoullos and Papageorgiou (1984) and Gupta and Varga (1992). In this paper, we prove that the normal distribution can be characterized via one conditional normal distribution and another constant conditional variance. The gamma distribution is another commonly used continuous distribution. Since Lukacs (1955) characterized two independent non-degenerate positive ran- dom variables X and Y to be gamma distributed by the independence of XIY and X + Y, many papers have been devoted to the problem of characterizing the gamma distribution. Wesolowski (1989) gave a characterization of the gamma distribu- tion via the first two conditional moments under the condition that X and Y are square integrable. In Sec. 3, we give the same characterization under different conditions. Some related results are given in Nguyen et al. (1996). 2. Characterization of the Normal Distribution In this section, given two random variables X and Y, let fA). fy(') denote respec- tively their density functions and let fYIX(Ylx) denote the conditional density of * Address for correspondence: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403-0221, USA. E-mail: gupta@bgnet.bgsu.edu. 59