This paper focuses on the development of analytical methods for uncertainty quantification of the models obtained by the Eigensystem Realization Algorithm (ERA) to quantify the effect of noise in the input-output experimental data. Starting from first principles, analytical expressions are presented for the probability distribution of eigenvalues of the Hankel matrix by application of standard results in random matrix theory. This result naturally leads to a probabilistic method for model order determination (reduction). By application of further results from the theory of random matrices, we develop analytical expressions for the marginal probability density of eigenvalues. Numerical examples illustrate the applications of ideas presented in the paper.