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Probability and Mathematics Statistics

**Course Code****:** B82003Y

**Course Name****: **Probability and Mathematics Statistics

**Credits****:** 4.0

**Level****:** Undergraduate

**Pre-requisite****: **Calculus and Analysis, Linear Algebra

**Lecture Time****: **80 hours

**Course Description**

This course is designed to provide the undergraduate students with a solid background and understanding of the basic results and methods in probability and mathematics statistics needed in more advanced research in related subjects. This course introduces basic concepts, tools, theories and primitive applications of probability and mathematics statistics including random variable and density, mathematical expectation and L-S integral, law of large numbers and central limit, parameter estimation and hypothesis testing, Bayesian estimation and regression analysis. This course intends to help students with major in science, engineering, and other related fields to develop their computing skills of probability statistics and advanced ability to solve practical problems with mathematics.

**Course Evaluation: **

The course grade will be determined by Homework (20%) and Final exam (80%). A weekly homework will be given during every week of the class, the homework will be graded and scores will count for 20% of the total. At the end of the class, there will be final exams, which will count for 80%. The full score of this course is 100.

**Topics and Schedule **

Chapter 1: Basic Concepts of Probability (12 hrs)

1.1. Sample space and random event

1.2. The definition of probability and its properties

1.3. Classical models of probability

1.4. Conditional probability, whole probability formula and Bayesian formula

1.5. Independence

Chapter 2: Random variables and their distribution (12 hrs)

2.1. Random variables

2.2. Discrete random variables and their distribution

2.3. Distribution function

2.4. Random variables of continuous and their distribution density function

2.5. Distribution of random variables function

2.6. Riemann-Stieltjes integral

2.7. Impulse function and the density function of discrete random variables

Chapter 3: Multi-dimensional random variables and their distributions (10 hrs)

3.1. Multi-dimensional random variables

3.2. Marginal distribution

3.3. Conditional distribution

3.4. Independence of Random variables

3.5. Distribution of multi-dimensional random variables function

3.6. Convolution of distributions function

Chapter 4: Numerical characteristics of random variables (10 hrs)

4.1. Mathematical expectation

4.2. Variance

4.3. Moment, Covariance and Correlation coefficient

4.4. Generating function

4.5. Conditional mathematical expectation

4.6. Applications of Conditional mathematical expectation

Chapter 5: Law of large numbers and Central limit theorem (8 hrs)

5.1. Sequence of random variables and its convergence analysis

5.2. Chebyshev inequality and law of large numbers

5.3. Characteristic function

5.4. Central limit theorem

Chapter 6: Sample and Sampling distribution (6 hrs)

6.1. Introduce of Mathematical statistics

6.2. Population, Individuality, Sample and Statistic

6.3. Sampling distribution

6.4. Distributions of order statistics

Chapter 7: Parameter estimation (8 hrs)

7.1. Point estimation and unbiased estimation

7.2. Moment estimation, consistency and sufficiency of estimation

7.3. Maximum likelihood estimation and EM algorithms

7.4. Minimum variance estimation

7.5. Interval estimation

Chapter 8: Hypothesis testing (8 hrs)

8.1. Hypothesis testing methods and Potential problems

8.2. One-tailed test and Two-tailed test

8.3. U-test and t-test

8.4. Confidence interval

Chapter 9: Basic of the Bayes statistics (4 hrs)

9.1. Bayesian estimation

9.2. Bayesian analysis

9.3. Statistical Decision

**Textbook**

[1] Zhou Sheng, Shiqian Xie, Chengyi Pan, Probability and Mathematics Statistics, 4th Edition, Higher Education Press, 2011.

**References**

[1] Mao shisong, cheng yiming and Pu xiaolong, Introduction to Probability and Mathematics Statistics, 2th Edition, Higher Education Press, 2011.

[2] A. N. Shiryaev, Probability-1, 3rd Edition, Springer Press, 2004.