The present research concerns the dynamical analysis of mistuned rotating bladed-disks for which nonlinear geometrical effects exist. The present methodology requires the construction of an adapted reduced-order basis from which a nonlinear reduced-order model is constructed. The mistuning phenomenon is taken into account by considering a nonparametric probabilistic approach based on the information theory. In the present context the uncertainty is introduced by replacing the reduced-order basis with a stochastic reduced-order basis (SROB). This latter one is obtained by using a new nonparametric probabilistic approach of model-form uncertainties so that each realization of the SROB respects some mathematical properties linked to the available information under constraints concerning the specified boundary conditions and the usual orthogonality properties. With such strategy, the computational effort is focused on the stochastic nonlinear reduced internal forces and the related tangential operator which are explicitly constructed using the SROB combined with the finite element method. The numerical application is a rotating mistuned bladed-disk subjected to a load for which geometrical nonlinearities effects occur. The uncertainty propagation in this nonlinear dynamical system is then analyzed.