【摘要】 针对正态分布概率密度函数的判定及数字特征关联问题展开研究。鉴于正态分布概率密度函数的原函数非初等函数,其统计特性需依赖反常积分求解,导致应用受限,而二维正态分布的判定尤为复杂。从二重积分切入,结合正定矩阵性质,推导得出一维与二维正态分布概率密度函数的关键判定条件,提出基于函数参数的快速判别方法。同时,深入剖析了正态分布核心数字特征与概率密度函数参数系数的定量关系,并通过应用实例验证了结论的有效性。研究成果为正态分布的快速识别提供了理论依据,可直接应用于统计学领域以提升抽样分布效率,为参数换算模型的构建奠定基础。更多还原

【Abstract】 This paper investigates the identification of the probability density function of normal distribution and the correlation of its numerical characteristics. Since the original function of the normal distribution probability density function is not an elementary function, its statistical properties must be determined through improper integrals, which limits its practical applications. Moreover, the determination of two-dimensional normal distribution is particularly complex. This study approaches from double integrals and, combined with the properties of positive definite matrices, derives the key criteria for determining one-dimensional and two-dimensional normal distribution probability density functions and proposes a rapid determination method based on function parameters. At the same time, it delves into the quantitative relationship between core digital characteristics of the normal distribution and the coefficients of the probability density function parameters and the validity of the conclusions is verified through application examples. The research results provide a theoretical basis for the rapid identification of normal distributions and can be directly applied in the field of statistics to improve sampling distribution efficiency and lay a foundation for the construction of parameter conversion models.更多还原