Probability

Course Code: B12005Y

Course Name: Probability

Credits: 4.0

Pre-requisite: Calculus, Linear Algebra, Complex Analysis

Lecture Time: 80 periods

Course Description

As a required basic course following calculus and linear algebra, this course is a mathematics course that studies the statistical law of random phenomena in the real world. Through this course, students can master the basic knowledge of probability theory and basic computation skills. Meanwhile, students shall use stochastic mathematics methods to solve problems. This course can also provide essential mathematical knowledge for the study of mathematical statistics, stochastic processes, and other relevant courses and expand modern mathematics knowledge.

Topics

1. Random Events and Probability

1.1.   Random phenomena and statistical regularity

1.2.   Sample space, random event and its operation

1.3.   Classical probability, geometric probability

1.4. Probability space, axiomatic structure of probability

2. Conditional Probability and Independence

2.1.  Conditional probability and multiplication formula

2.2.  Full probability formula and Bayes formula

2.3.  Independent events

2.4.  Independent repeat test and Bernoulli test

3. Random Variable and Its Distribution

3.1.   Random variables and their distributions(discrete, continuous)

3.2.   Multivariate random variables and their distributions (discrete, continuous)

3.3.   Conditional distribution and the independence of random variables

3.4.Function of random variable and its distribution

4. Numerical Characteristics of Random Variables

4.1.  Expectation and variance, moment

4.2.  Covariance and correlation coefficient

4.3.  Entropy and information quantity

4.4. Conditional expectation and optimal forecast

4.5. Generating function, Laplace transform, moment generating function, characteristic function

5. Limit Theorems

5.1.   Convergence of random variables

5.2.   Weak law of large numbers, strong law of large numbers

5.3.   The central limit theorem

6. Preliminary Stochastic Processes

6.1.  Stochastic processes and their finite dimensional family of distributions

6.2.   Independent increment process, stationary increment process, two order moment process, orthogonal increment process

6.3.  Gauss process, Brown motion, Poisson process

6.4 Preliminary Markov chain with discrete time

6.5.Preliminary stationary process and its ergodic theory

7. Stochastic Simulation

7.1. Monte Carlo method and random number generation

7.2. Random variables simulation (discrete type, continuous type)

7.3. Markov chain simulation

7.4. Approximate calculation of integrals

Textbook

[1] Sheldon M.Ross, A First Course in Probability, Ninth Edition, Pearson, 2012

[2] Sheldon M.Ross, A First Course in Probability, Ninth Edition, China Machine Press, 2014 (Chinese Version)

References

[1] William Feller, An Introduction to Probability Theory and Its Application, Third Edition, Posts and Telecom Press, 2014

[2] Chen Jiading, Zheng Zhongguo, Probability and Statistics, Peking University Press, 2007

[3] Xiao Shutie, Qian Minping, Ye Jun, College Mathematics - stochastic mathematics, Second Edition, Higher Education Press, 2004

[4] Miao Baiqi, Hu Taizhong, A Course In Probability Theory , Second Edition, University of Chinese Science and Technology of China Press, 2009