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中文摘要：
利用模糊推理建立了一种基于输入-输出数据构造联合概率密度函数的方法。首先，将一组单输入一单输出数据转换成模糊推理规则，通过选择适当的模糊蕴涵算子生成模糊关系，再利用这种模糊关系求出二维随机变量的联合概率密度函数。当将模糊蕴涵分别取为Larsen蕴涵和Mamdani蕴涵时，分别得到了两种具体的概率密度函数（称之为Larsen分布和Mamdani分布）。其次，分别研究这两种概率分布的边缘分布和数字特征，指出这两种概率分布有相同的数学期望，有几乎相同的方差和协方差。从而进一步揭示模糊系统的概率论意义。
英文摘要：
In this paper, based on a set data of input-output, a method of constructing joint probability density functions by using fuzzy inference is proposed. Firstly, a set data of input-output is transferred into fuzzy inference rules and the joint probability density functions for some binary random variable are derived by choosing fuzzy implications and building fuzzy relations. When Larsen implication and Mamdani implication are chosen, two kinds of probability density functions, which are called Larsen distribution and Mamdani distribution respectively, are obtained. Secondly, the marginal density functions and numerical characteristics of the Larsen distribution and the Mamdani distribution are discussed respectively. It is pointed that the two probability distributions have the same mathematical expections and have the almost same variances and covariances. These discussions further reveal the probability consideration of fuzzy systems.
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