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In this paper, a unified uncertainty analysis method based on the extension universal generating function is proposed for engineering problems described by the mixture of random, interval and p-box variables. In the method, the univariate approximation approach is extended for mixed variables, and then the performance function is divided into three parts. Traditional universal generating function can only model the case that all variables exist in system are random variables. However, in order to calculate the bounds of system probability of failure under mixed variables, the mixed universal generating function is developed to extend the traditional universal generating function. The optimization models based on the mixed universal generating function are presented to calculate system probability of failure under mixed variables. An engineering example is used to validate the proposed method.