Turbulent dynamical systems are ubiquitous in applications in contemporary science and engineering where statistical ensemble prediction and real-time filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scientific phenomena. Here a class of statistically exactly solvable non-Gaussian test models are introduced where a generalized Feynman-Kac formulation reduces the exact behavior of conditional statistical moments to the solution of inhomogeneous Fokker-Planck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and combined with information theory to address two important issues in contemporary statistical prediction of turbulent dynamical systems: coarse-grained ensemble prediction in a perfect model and improving long range forecasting in imperfect models. Besides, the tool developed here is applied to filter the turbulent signal with different type of prediciton models. Here both Nonlinear Extended Kalman Filter and Non-Gaussian Filter are studied. The models discussed here should be useful for many other applications and algorithms for real time prediction and state estimation.