Characterizations of Continuous Binomial and Negative Binomial Mixtures

Characterizations are given for binomial (n, ?) and negative binomial (k, ?) mixtures in terms of the regression function of the mixing random variable Y on the mixture variable X. In both cases Y is the parameter ? and the mixing distribution is a continuous distribution on (0, 1) and on (0, ∞) respectively. Biological applications are indicated.