The multivariate Selberg probability bound provides a useful tool in distributionally robust optimization when the objective depends on the probability of a random vector with fixed second-order moments falling outside the ellipsoid that is not necessarily centered at the mean. The main result presented in the paper establishes an analytical representation of the exact probability bound in the case when the mean is unknown but bounded in the Euclidean norm while the covariance is a fixed matrix of an arbitrary form. The paper demonstrates how this result of stochastic optimization can be effectively applied to distributionally robust estimation problems.