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There are two custom ways for predicting RNA secondary structures: minimizing the free energy of a conformation according to a thermodynamic model and maximizing the probability of a folding according to a stochastic model. In most cases, stochastic grammars are used for the latter alternative applying the maximum likelihood principle for determining a grammar's probabilities. In this paper, building on such a stochastic model, we will analyze the expected minimum free energy of an RNA molecule according to Turner's energy rules. Even if the parameters of our grammar are chosen with respect to structural properties of native molecules only (and therefore, independent of molecules' free energy), we prove formulae for the expected minimum free energy and the corresponding variance as functions of the molecule's size which perfectly fit the native behavior of free energies. This gives proof for a high quality of our stochastic model making it a handy tool for further investigations. In fact, the stochastic model for RNA secondary structures presented in this work has, for example, been used as the basis of a new algorithm for the (nonuniform) generation of random RNA secondary structures.