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课程教学大纲选节(中译英)

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发表于 2020-2-21 11:39:58 |显示全部楼层
本帖最后由 Fiona 于 2020-2-21 11:38 编辑

课程教学大纲选节

2.1 课程简介
概率论是数学系,金融工程各专业必修的基础课之一. 随机现象广泛存在于人类活动的各个方面. 随着科技的不断进步,人们对定量分析随机现象的规律性提出了越来越高的要求.概率论就是研究和揭示随机现象中统计规律性的数学分支, 不同于其他研究必然现象的数学分支,它是一门应用性很强的数学学科,广泛地应用于金融、保险,证券,工程技术和自然学科中,是各学科中分析与解决随机问题的基本工具. 通过本课程的学习,使学生比较系统地获得概率论的基础知识,使学生初步掌握处理随机现象的基本思想与方法,具备分析和处理带有随机性数据的理论基础,为后续数理统计等相关课程的学习打下必要的基础.
2.2 课程目标具备扎实的数学理论基础;善于用随机思维分析不确定现象;能利用概率方法解决现实中的随机问题.
2.3 课程目标与毕业要求的支撑关系
通过较严格的随机思维的训练,掌握概率的随机思想方法具备在现实情境中发现问题,提出问题,分析问题,解决问题的能力;从中推理了解数学学科的理论及应用的新发展,具有一定的创新意识和创新能力对事物进行观察、比较、分析、综合、抽象、概括、判断、推理的能力,采用科学的逻辑方法,准确而有条理地表达自己思维过程的能力.
三、课程教学内容与学时分配
第一章随机事件及其概率
1. 学习目的. 了解随机试验,随机事件的概念;掌握概率的几种定义;熟练利用性质,定理计算概率.
2. 课程内容
(1). 随机现象,随机事件,基本事件,样本空间基本概念.
(2). 随机事件的关系及运算.
(3). 古典概率.
(4). 几何概率及统计概率.
(5). 公理化概率及概率的性质.
(6). 条件概率,全概率公式,贝叶斯公式.
(7). 事件的独立性,贝努里概型.
3. 教学重点与难点
教学的重点是利用定义,性质,定理计算概率.难点是选择合适的方法计算概率.
4. 思考与练习
教材中第一章的习题.
第二章  随机变量及其概率
1. 学习目的. 掌握随机变量分布函数的定义,意义和计算方法.熟悉常见的离散型,连续型随机变量.
2. 课程内容
(1). 一维随机变量及其分布函数.
(2). 一维离散型随机变量及其分布,分布律, 0-1分布,二项分布,泊松分布,几何分布.
(3). 一维连续型随机变量及其分布,密度函数,均匀分布,指数分布,正态分布.
  (4). 一维随机变量函数的分布.
  (5). 二维随机变量的联合分布,边缘分布,条件分布.
  (6). 二维离散型随机变量的联合分布,边缘分布,条件分布,独立性.
  (7). 二维连续型随机变量的联合分布,边缘分布,条件分布,独立性.
  (8). 二维均匀分布,二维正态分布.
  (9). 二维随机变量函数的分布.
3. 重点与难点
    重点是掌握随机变量分布函数的计算.难点是理解随机变量分布函数的意义 积分计算中积分上下限的确定.
4. 练习与思考
教材第二章的习题.
第三章随机变量的数字特征
1. 学习目的. 理解随机变量数学期望,方差,矩,协方差,相关系数的含义并能够计算. 理解随机变量特征函数的定义意义. 掌握特征函数的性质.
2. 课程内容
(1). 随机变量的数学期望及其性质.
(2). 随机变量的方差及其性质.
(3). 随机变量矩、协方差、相关系数及其性质.
(4). 随机变量的特征函数及其性质.
3. 教学重点与难点
教学重点是掌握各个数字特征的定义和性质. 难点每个数字特征的意义和特征函数性质的灵活应用.
4. 练习与思考
教材第三章的习题.
II.Course Objectives and Learning Expectations
2.1Course Description
Probability Theory is one of the basic courses required for the majors of Mathematics and Financial Engineering. As random phenomena are widely present in all aspects of human activities and with the continuous advancement of science and technology, people are requiring more for the regularity of quantitative analysis of random phenomena. Probability Theory is a branch of mathematics that studies and reveals statistical regularity in random phenomena. Different from other mathematical branches that study inevitable phenomena, probability theory is a highly applied mathematical discipline widely used in finance, insurance, securities, engineering technology and natural sciences. It is the basic tool for analyzing and solving random problems in various disciplines. Through the study of this course, students can systematically acquire the basic knowledge of probability theory, grasp the basic ideas of dealing with random phenomena, acquire the theoretical basis for analyzing and processing random data, and lay the necessary foundation for the follow-up of mathematical statistics and other related courses.
2.2 Course Objectives: Lay a solid foundation of mathematical theory, be good at analyzing uncertain phenomena with random thinking, use probability methods to solve real random problems.
2.3 Supporting of Course Objectives for Graduation Requirements
Through the strict training of random thinking, a student can master the random thought method of probability, know how to find problems, ask questions, analyze problems and solve problems in the real situation, understand the new development of the theory and application of mathematics from the reasoning, apply innovation to observe, compare, analyze, synthesize, abstract, generalize, judge, and infer things, adopt scientific logic methods, and accurately and systematically express the ability of their thinking process.

III.Course Content and Class Hours
Chapter I Random Events and Their Probabilities
1.Learning Objectives. Understand the concept of random experiments, random events; master several definitions of probability; calculate probability with theorem based on nature.
2.Course Content
A.Basic concepts, such as random phenomena, random events, basic events, and sample space.
B.The relationship and operation of random events.
C.Classical probability.
D.Geometric probability and statistical probability.
E.The axiomatization probability and the nature of probability.
F.Conditional probability, full probability formula, Bayesian formula.
G.The independence of the event, the Bernoulli profile.
3.Key Points and Difficult Points in Teaching
The key point of teaching is to use the definition, nature, and theorem to calculate the probability. The difficult point is to choose the appropriate method to calculate the probability.
4.Thinking and Practicing
The exercises in Chapter I of the textbook.
Chapter II Random Variables and Their Probabilities
1.Learning Objectives. Master the definition, meaning and calculation method of the distribution function of random variables. Familiar with common discrete and continuous random variables.
2.Course Content
A.One-dimensional random variables and their distribution functions.
B.One-dimensional discrete random variables and their distribution, distribution law, 0-1 distribution, binomial distribution, Poisson distribution, geometric distribution.
C. One-dimensional continuous random variables and their distribution, density function, uniform distribution, exponential distribution, normal distribution.
D.Distribution of one-dimensional random variable functions.
E.Joint distribution of two-dimensional random variables, edge distribution, conditional distribution.
F.Joint distribution of two-dimensional discrete random variables, edge distribution, conditional distribution, independence.
G.Joint distribution of two-dimensional continuous random variables, edge distribution, conditional distribution, independence.
H.Two-dimensional uniform distribution, two-dimensional normal distribution.
I.Distribution of two-dimensional random variable functions.
3.Key Points and Difficult Points in Teaching
The key point is to master the calculation of the distribution function of random variables. The difficult point is to understand the meaning of the distribution function of random variables, and the determination of the upper and lower limits of the integral in the integral calculation.
4.Thinking and Practicing
The exercises in Chapter II of the textbook.
Chapter III Digital Features of Random Variables
1.Learning Objectives. Know how to define and calculate the mathematical expectation, variance, moment, covariance, correlation coefficient of random variables, understand the definition and significance of the characteristic function of random variables and master the nature of the characteristic function.
2.Course Content
A.The mathematical expectation of random variables and their properties.
B.The variance of random variables and their properties.
C.The moment, covariance, correlation coefficient and properties of random variables.
D.The characteristic function of random variables and their properties.
3.Key Points and Difficult Points in Teaching
The key point of teaching is to master the definition and nature of each digital feature. The difficult points include the flexible application of the meaning of each digital feature and the nature of the characteristic function.
4.Thinking and Practicing
The exercises in Chapter III of the textbook.


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