The procedures of the identification of probability distributions for K(≥1) random objects, each having one from the known set of M distributions, are studied. K sequences of discrete independent random variables represent results of N observations of each of these objects. The exponential decrease of test error probabilities is considered. The reliability matrices of logarithmically asymptotically optimal tests are investigated for some models. These models are determined by conditions of dependence or independence of objects and by the formulation of an identification problem. The optimal subsets of reliabilities which may be given beforehand and conditions of positiveness of all of the reliabilities are investigated.