矩量母函数是概率论研究中的一项重要内容，本论文首先介绍了矩量母 函数的简单性质如卷积公式和其与随机变量的矩的关系，矩量母函数能唯一 确定分布函数的条件，随机变量的矩量母函数收敛与其分布收敛的关系，之 后根据其定义域固定的特点引出了拉普拉斯变换与非负随机变量反演公式使 得可以根据随机变量的矩量母函数反推出随机变量的分布函数。最后研究了 随机变量前 n 项和较大时与 n 倍均值的偏离程度，给出了明确的衰减速率公 式并证明其正确性，其对于稀有事件的概率有较好的指数型估计。 本文的创新点主要有： • 研究了目前国内概率论教材中提及较少的矩量母函数的详细性质； • 详细介绍了除了计算随机变量各阶矩量以外矩量母函数的其他应用。

Moment generating function is an important element in the study of probability theory, this thesis first introduces the simple properties of the moment generating function such as the convolution formula and its relationship with the moments of the random variable, moment generating function can uniquely determine the conditions of the distribution function, the relation between the convergence of moment generating function and the convergence of distribution, and then according to the characteristics of the definition of the domain of a fixed Laplace transform and the inverse of the non-negative random variable formula. Then the Laplace transform and the inverse formula for non-negative random variables are introduced according to the characteristics of its fixed definition domain, which makes it possible to invert the distribution function of a random variable according to the moment generating function of the random variable. Finally, the degree of deviation of the first n terms of a random variable from the n-fold mean is investigated, and an explicit formula for the decay rate is given and proved to be correct, which provides a good exponential estimate of the probability of a rare event.