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In this study, a high-gain non-linear observer-based fault diagnosis (FD) approach is proposed for a class of non-linear uncertain systems with measurable output probability density functions (PDFs). The objective of the presented FD algorithm is to use the measurable output PDFs and the input of the system to construct an exponential observer-based residual generator such that the fault can be detected and diagnosed. The main result is given in a constructive manner by developing a novel non-linear observer, without resort to any linearisation. By a coordinates transformation, the design of the proposed observer does not need to solve any kind of linear matrix inequalities and its expression is explicitly given. The exponential convergence of the errors in the presence of uncertainties is proved to guarantee the fastness of the proposed FD scheme by employing a class of quadratic Lyapunov functions. Furthermore, the bound of the estimation errors in the presence of faults is minimised by appropriately choosing the parameters of the presented observer. Finally a simulation example is given to illustrate the effectiveness of the proposed FD method.