University of Chinese Academy of Sciences

Course Code: B62001H

Course Name: Probability and Mathematics Statistics

Credits: 4.0

Level: Undergraduate

Pre-requisite: Advanced Mathematics, Linear Algebra

Lecture Time: 80 hours

Course Description

This course introduces fundamental conceptions, theories and primitive applications of probability and mathematics statistics including random variable, random process, law of large numbers and central limit, parameter estimation and hypothesis testing, 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.

Topics and Schedule

1. Introduction (2 hrs)

1.1.   Concepts of probability research

1.2.   Development and application of probability and mathematics statistics

1. Random event and Probability (8 hrs)

2.1.   Random phenomena, trial and event

2.2.   Set theory and Sample space

2.3.   Definition and Basic properties of probability

2.4.   Basic operation and Conditional probability

2.5.   Axioms of Probability and Bayes theorem

2.6. Statistical Independence

1. Random variables and Probability distributions (12 hrs)

3.1.   Discrete random variables
(Zero-one distribution, Binomial distribution, Poisson distribution)

3.2.   Continuous random variables
(1D Normal distribution, Uniform distribution, Exponential distribution)

3.3.   Review and Problems for Chapter 2 and 3

1. Multi-dimensional random variables (6 hrs)

4.1.   Distribution function of Bivariate random variable

4.2.   Marginal distribution and Conditional distribution

4.3.   Independence of random variables

4.4.   Distribution functions of random variables

1. Numerical characteristics of random variables (10 hrs)

5.1.   Expectations and Variance

5.2.   Numerical characteristics of some typical distributions

5.3.   Moment, Covariance and Correlation coefficient

1. Markov chain and Simple stochastic process (8 hrs)

6.1. Transition probability, State, Chain

6.2. General Markov process

6.3. Continuous-time Markov process
(Poisson point process, Birth-death process, Simple queuing theory)

1. Law of large numbers and Central limit theorem (8 hrs)

7.1. Chebyshev inequality and Law of large numbers

7.2. Central limit theorem of independent identically distributed

7.3. Review and problems of Chapter 4-7

1. Introduce of Mathematical statistics (6 hrs)

8.1. Population, Individuality, Sample and Statistic

8.2. Sampling distribution
              (Chi-square distribution, T distribution, F distribution)

8.3. General sampling distribution of Normal population

1. Parameter estimation (4 hrs)
   1. Point estimation, Moment estimation, Maximum likelihood estimation
   2. Unbiasedness, Consistency and Sufficiency of Estimation
   3. Interval estimation, Mean of Normal population, Confidence interval of Variance
2. Hypothesis testing (4 hrs)

10.1 Hypothesis testing methods and Potential problems

10.2 One-tailed test and Two-tailed test

10.3 U-test and t-test

1. Regression analysis (6 hrs)

11.1 Linear regression analysis

11.2 Least square methods

11.3 Review and problems of Chapter 7-11

12. Tests and Q-course (6 hrs)

Grading

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 middle and end of the class, there will be midterm and final exams, which will count for 20% and 60% respectively. The full score of this course is 100.

Textbook

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

[2]    William Filler,An introduction to Probability Theory And Its Applications, 3^rd Edition, People's Posts and Telecommunications Press, 2013

References

[1]   Rick Durrett, Probability: Theory and Examples(4th edition), Cambridge University Press, 2010

[2]   Jun Shao, Mathematical Statistics, Springer-Verlag, 2010