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This paper considers a new type of fault detection and diagnosis (FDD) problem for general stochastic systems. Different from the classical FDD problems, the measured information is the probability distribution of system output rather than the value of output. The objective is to find an observer-based residual by using the output distributions such that the fault can be detected and diagnosed. Square root B-spline expansions are applied to model the output probability density functions (PDFs) so that the concerned problem is transformed into a nonlinear FDD problem subject to the weight dynamical systems. An LMI-based solution is presented such that the estimation error system is stable and the fault can be detected through a threshold. Moreover, an adaptive fault diagnosis method is also provided to estimate the size of the fault.