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A large deviations approach is introduced, which calculates the probability density and outage probability of the multiple-input multiple-output (MIMO) mutual information, and is valid for large antenna numbers N . In contrast to previous asymptotic methods that only focused on the distribution close to its most probable value, this methodology obtains the full distribution, including its non-Gaussian tails. The resulting distribution interpolates between the Gaussian approximation for rates R close its mean and the asymptotic distribution for large signal-to-noise ratios (SNRs) ρ . For large enough N, this method provides the outage probability over the whole (R, ρ) parameter space. The presented analytic results agree very well with numerical simulations over a wide range of outage probabilities, even for small N. In addition, the outage probability thus obtained is more robust over a wide range of ρ and R than either the Gaussian or the large-ρ approximations, providing an attractive alternative in calculating the probability density of the MIMO mutual information. Interestingly, this method also yields the eigenvalue density constrained in the subset where the mutual information is fixed to R for given ρ. Quite remarkably, this eigenvalue density has the form of the Marčenko-Pastur distribution with square-root singularities.