Consider X and Y to be independent and non-degenerate random variables. If X and Y are non-negative then it follows from (6) that for all given values z of X + Y, the expected values of X and X2 are of the type λz and λ′ z2, respectively, if and only if X and Y have the gamma distributions with the same scale parameter. This is an improvement over Lukacs's result (3). Since the characterization problems for the Poisson, binomial, negative binomial and normal distributions resemble the corresponding problem for the gamma distribution it is quite reasonable to expect the characterizations similar to the above for these distributions. This is established in what follows.