We investigate existence and properties of discrete mixture representations $P_θ=\sum_{i\in E} w_θ(i) \, Q_i$ for a given family $P_θ$, $θ\inΘ$, of probability measures. The noncentral chi-squared distributions provide a classical example. We obtain existence results and results about geometric and statistical aspects of the problem, the latter including loss of Fisher information, Rao-Blackwellization, asymptotic efficiency and nonparametric maximum likelihood estimation of the mixing probabilities.