On some characterizations of population distributions

In this paper, earlier works of the present authors and a method due to Ano sov for solving certain intego-functional equations are combined to show th at the independence of the sample mean (X) over bar (n) and the Z(n)-statis tic characterizes the normal population, when the random samples are lid fr om a population having a continuous density function on R, and the sample s ize n greater than or equal to 3; obviously the sample standard deviation i s a Z(n)-statistic. Further, an important subclass of Z(n)-statistic with t he form of a linear combination Sigma (n)(i=1) a(i)X((i)) of order statisti cs is found, where a(1) less than or equal to...less than or equal to a(n), not all equal and Sigma (n)(i=1) a(i)=0, which includes Gini's mean differ ence and the sample range but not the sample standard deviation. Similar approach can be applied to prove that the independence of (X) over bar (n) and Z(n)/(X) over bar (n) characterizes the gamma distribution; obv iously the independence of sample mean and sample coefficient of variation characterizes the gamma distribution. The study of identifying Z(n) to more known statistics will be the future w ork.