3rd International Conference on Education, Management, Arts, Economics and Social Science (ICEMAESS 2015)
Teaching Research in Probability Theory and Mathematical Statistics
Modeling Idea
Li Zhang
College of mathematics and information, China West Normal University, Nanchong Sichuan, 637000, China
Keywords: Mathematical modeling idea, Probability theory, Probability and statistics, Mathematical statistics, Case teaching
Abstract. Integrating the idea and approach of mathematical modeling into the Probability Theory and Mathematical Statistics can provide new idea for the teaching process, make the relation between theory and practice more closers and motivate studentsrsquo; interest in learning. This paper discusses the necessity and feasibility to integrate mathematical modeling into this course and analyzes the whole teaching process based on teaching cases from different angles.
Introduction
Probability Theory and Mathematical Statistics is one of important core mathematics disciplines under the current background of modern information engineering and it is also the specific method for current social economy study and application. In traditional teaching method, the content of this discipline is boring and rigid in ideas, so it fails to let the students to be enthusiastic in learning. If we attempt to integrate mathematical modeling into the design of the course teaching, it can increase the applicability of this course and make it be more closer to professional background and studentsrsquo; life, not only letting students feel the charm of math, but also letting students improving themselves in active practice and exploration.
Brief Introduction Probability Theory and Mathematical Statistics
Probability Theory and Mathematical Statistics is a scientific discipline with math as its core to summarize statistical regularity based on random phenomenon. This discipline can reveal the inevitability and contingency of some incidents from the perspective of quantification and discuss the relation between the two. The study in the theory content and summarization of scientific methods based on abstract random incidents is helpful for cultivating studentsrsquo; logical thinking ability to solve practical problems.
Integration of mathematical modeling in the 'Probability Theory and Mathematical Statistics' is a more creative and new attempt for teaching, which is based on the characteristics of the actual problem and the law of human production and life and uses abstract methods to extract more specific quantitative mathematical problems for this course, utilizes mathematical thinking to solve and analyze the subjects. And the results can later be used to answer the real questions after its results being interpreted and verified. For example, the problems such as finance and investment, winning the lottery, birthdays and other probability problems in real life can be applied and answered, combing the idea of mathematical modeling with mathematical statistics courses. Some education scholars have pointed out that 'the introduction of the mathematical modeling with innovative ideas brings more freshness and creativity for probability theory and mathematical statistics course, if further its exploration, you#39;ll find its practical and far-reaching significance for the course. This is an outstanding attempt to train studentsrsquo; analytical skill in theoretical knowledge and consciousness to solve problems.
copy; 2016. The authors - Published by Atlantis Press
The Necessity and Feasibility of Integrating Mathematical Modelling Idea into Teaching of
Probability Theory and Mathematical Statistics
Necessity. First, the discipline of 'Probability Theory and Mathematical Statistics' is a comprehensive discipline that integrates rational thinking and observing trials, which bears very rich solving methods for mathematical problems such as the more common model method, construction method, transformation method, quantitative methods, and so on. Especially from the traditional classical concept of mathematical model to today#39;s more popular regression analysis models, mathematical modeling idea can always be integrated into courses of this discipline. Probability theory and mathematical statistics based on mathematical modeling puts more emphasis on the rational and thorough analysis on real data to identify some of the necessary data parameters and testing hypotheses in the courses through mathematical modeling. But in fact, some students were not able to use the basic statistical knowledge in mathematical modeling to analyze the real data, which caused that the regression model constructed by them did not achieve the desired results in forecasting time series, therefore introducing the idea of mathematical modeling into ' Probability Theory and Mathematical Statistics ' and making it comparable with the theoretical basis of the course based on practice is conducive for improving the value of the theory and practice of the course.
Feasibility. As it is mentioned above, the 'Probability Theory and Mathematical Statistics” is quite good at processing quantitative indicators data of random phenomena. Its theoretical approach has been recognized by almost all walks of life in society. For example national welfare lottery, environmental pollution control, corporate investment strategies, and social security and so on all choose to use the knowledge of this discipline. The first step of their practices is to conduct quantitative sampling on the issues and solve the problems after making reasonable examination and judgment, so it is feasible to gradually deepen mathematical modeling idea in this discipline. In addition, the integratio
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概率论与数理统计建模思想在教学研究中的应用
关键词：数学建模思想，概率论，概率与统计，数理统计，案例教学
概要：将数学建模的思想和方法与概率论、数理统计相结合，可以为教学过程提供新的思路，使理论与实践的关系更加紧密，激发学生的学习兴趣。本文论述了将数学建模融入本课程的必要性和可行性，并从不同的角度对基于教学案例的整个教学过程进行了分析。
介绍：概率论与数理统计是现代信息工程背景下重要的核心数学学科之一，也是当前社会经济研究与应用的具体方法。在传统的教学方法中，这门学科的内容枯燥，思想僵化，不利于学生的学习热情。如果我们尝试将数学建模融入课程教学的设计中，可以提高本课程的适用性，使其更贴近专业背景和学生的生活，不仅让学生感受到数学的魅力，而且让学生在积极的实践和探索中提高自己。
数学建模思想融入概率论与数理统计教学的必要性和可行性
必要性。第一，“概率论与数理统计”学科是一门集理性思维与观察实验为一体的综合性学科，对较为常见的模型法、构造法、变换法、定量法等数学问题具有十分丰富的求解方法。特别是从传统的经典数学模型概念到当今比较流行的回归分析模型，数学建模思想总能融入到这门学科的课程中。基于数学建模的概率论和数理统计更注重对真实数据的理性和深入分析，通过数学建模识别课程中一些必要的数据参数和测试假设。
可行性。如前所述，“概率论与数理统计”对于处理随机现象的定量指标数据是相当好的。其理论方法已被社会各界所公认。例如国家福利彩票、环境污染控制、企业投资策略、社会保障等都选择运用这门学科的知识。实践的第一步是对问题进行定量抽样，经过合理的审查和判断，解决问题，逐步深化本学科的数学建模思想是可行的。
经典概率问题的实例分析。生日问题是经典概率问题的经典例子之一。通过对经典概率论相关知识的教学和分析，加深学生对概率论理论知识的理解。例如，一个班有50个学生，其中至少有两个学生有相同的生日。在这种情况下，在“概率论和数理统计”课程中，假设一个班级总共有100人，公历中所有学生的出生日期都记录在匿名笔记上并上交。计算机随机抽取50组学生，同一测试在不同班级进行，100人连续8年，实验数据如下，如表1所示。
表1。至少两名同一生日的同学在连续八年的班级测试的概率
|
Item |
Class |
|||||||
|
the time of at least two students with the same birthday |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
34 |
37 |
35 |
38 |
36 |
34 |
39 |
35 |
|
从表1可以看出，至少两个学生有相同生日的概率高达71%。可根据经典概率计算推导：由此产生了一个新的问题：如果我们随机抽取50组学生的生日笔记，让学生回答同一个生日的所有学生的概率。把这个问题留给学生是为了进一步加深他们对经典概率知识的理解和灵活运用。另一方面，我们也可以通过抽签来考察学生在课堂上对概率论的理解程度，比如让学生考虑抽签的概率在最初和后来是否相同。
经济效益问题分析。概率论在现实生活中也很常见。它可以简单地节约复杂的问题。本文提出了几个这样的经济问题。
自然灾害预测
自然灾害预测是一项非常重要的工作，它直接影响到人们的生活和生产的效率和安全。假设有A、B、C、D四种方法可以用来预测可采用的自然灾害。如果这四种方法都单独用于预测，则费用如表2所示。
表2。四种自然灾害预报方法的成功概率统计以及他们的开支
|
Item |
Forecast Method |
|||
|
A |
B |
C |
D |
|
|
probability of success(expense/10,000 RMB) |
0.8 |
0.7 |
0.6 |
0.5 |
|
80 |
50 |
20 |
5 |
|
在此预测中，您可以使用某种单独的预测方法，也可以组合多种预测方法。如果预测成本的支出上限为1亿美元，您应该考虑哪种方法在预测自然灾害方面最有可能成功。因此，下面计算出三个完全独立事件的概率，并得出三个预测方案，而预测组合方法的选择是：A; A B; B C D,如表3所示。
表3。结合预测方法预测项目和预测成功率
|
Item |
Scheme |
||
|
A |
A B |
B C D |
|
|
probability of success |
0.8 |
1-（1-0.8）（1-0.7）=0.94 |
1-（1-0.7）（1-0.6）（1-0.5）=0.94 |
在基于联合预测方法建立新的预测方法后，发现其概率为0.94，高于选择某一特定方案的概率。教师解决和分析问题后，也可以要求学生对自编数据进行概率统计和数理统计，以便进一步完善和总结自编问题的经验。它的目的是通过联合预测方法获得更高的预测成功率。通过上述问题可以看出，概率论和数理统计在运用市场营销、人力资源管理等经济问题上是相当有效的，可以作为提出问题的材料。
基于中心极限定理的概率论案例分析-寿险问题
在“概率论与数理统计”课程中，中心极限定理的阐述与分析可以从数学建模思想中得到。它也是概率论中最关键的知识。运用实例分析法促进中心极限定理在教学中的应用，也可以激发学生学习概率论的兴趣。以下以某寿险公司的相关保险内容为例进行分析。
10万人参加城市人寿保险公司的养老保险，平均每人每年缴纳保险费3000元。投保人在一年内死亡的，保险公司应当赔偿受益人15000元。如果当地老年人死亡率为0.05，保险公司在老年人寿险业务中的年损失概率是多少？
假设a为该问题中被保险老年人每年的死亡人数，其中二项分布理论可用于使一项原则服从二项分布b（100000,0.05），np为100，np（1-p）=99。
保险公司发生损失的，必须符合下列条件：
3000100000-1000A＜0
If Agt;3000, then it is:
p（Agt;3000）=1-p(0＜A＜3000) asymp; 1-[]=2-2asymp;0
通过以上的推理和计算，运用数学建模思想和中心极限定理来解决概率问题，可以确定保险公司盈亏概率的答案，从而加深学生对学习内容的印象，也可以拓宽学生的思路。提高学生的实际计算能力，解决日常生活中的一些实际问题。
鉴于“概率论与数理统计”中涉及的问题处理数据的计算负担较大，因此教师在举例说明数据简化和内容分析时，应尽量结合计算机统计软件，采用多媒体方式演示SAS、SPSS和d.为学生提供的其他软件功能，优化一些学生的统计方法和统计模型，也可以明确数据处理和统计的步骤和结果。同时，在课堂教学中，教师应引导学生将数学建模思想结合起来，思考解决某些概率论问题的方法，并充分利用这些数据拟合、统计推理等相关实验，提高学生的应用意识和能力。概率论与数理统计课程
Teaching Research in Probability Theory and Mathematical Statistics
College of mathematics and information, China West Normal University, Nanchong Sichuan, 637000, China
Keywords: Mathematical modeling idea, Probability theory, Probability and statistics, Mathematical statistics, Case teaching
Abstract. Integrating the idea and approach of mathematical modeling into the Probability Theory and Mathematical Statistics can provide new idea for the teaching process, make the relation between theory and practice more closers and motivate studentsrsquo; interest in learning. This paper discusses the necessity and feasibility to integrate mathematical modeling into this course and analyzes the whole teaching process based on teaching cases from different angles.
Introduction
Probability Theory and Mathematical Statistics is one of important core mathematics disciplines under the current background of modern information engineering and it is also the specific method for current social economy study and application. In traditional teaching method, the content of this discipline is boring and rigid in ideas, so it fails to let the students to be enthusiastic in learning. If we attempt to integrate mathematical modeling into the design of the course teaching, it can increase the applicability of this course and make it be more closer to professional background and studentsrsquo; life, not only letting students feel the charm of math, but also letting students improving themselves in active practice and exploration.
The Necessity and Feasibility of Integrating Mathematical Modelling Idea into Teaching of
Probability Theory and Mathematical Statistics
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