授課教師：蔡碧紋  

（英文）：Introduction to Probability

一、教學目標：

本課程介紹機率論的基本概念和基本極限理論，並引導學生發掘機率的奧秘。

二、教材內容及進度規劃：

書目: Ross, S. (2009) A First Course in Probability 8th ed. Pearson Prentice Hall (華泰書局代理);

考古題：

Syllabus

1. Combinatorial Analysis

                        (a) Counting techniques: multiplication principle, permutations, combinations,

                        (b) Multinomial Coefficients
 

2. Probability: Set theory

                        (a) Probability set up: sample space, event

                        (b) axioms and some propositions

                        (c) What is probability? equally likely, relative frequency, a measure of belief. 

3. Conditional probability and Independence

                        (a) Conditional probability

                        (b) Bayes’ Formula

                        (c) Independent Events
 

4. Random variables

                        (a) What is random variables

                        (b) Cumulative distribution function

                        (c) Discrete R.V.s

                        (d) The Mean, variances and standard deviation

                        (e) Some discrete distributions:
                    Bernoulli trials, Binomial Distribution, Poisson Distribution
                    Geometric Distribution, Negative Binomial Distribution,
                    Hypergeometric Distribution,

                        (f) Moment generating functions (7.7)

                        (g) Poisson approximation to the Binomial

5. Continuous Distribution

                        (a) Continuous-type data

                        (b) Some continuous distribution: Uniform Distribution,

                    Mid-term I

                        Exponential Distribution, Normal Distribution

                        (c) Gamma and Chi-square Distribution

                        (d) Normal approximation to the binomial

                        (e) Distributions of functions of a random variable: Normal → Chi-square Distribution, Lognormal
   

                1 hour quiz                    

6. Jointly Distributed R.V.s

                        (a) Joint Distribution

                        (b) Independent r.v.s

                        (c) Sums of Independent r.v.s

                        (d) Conditional Distribution

                        (e) Joint Distribution of Functions of Random Variables

 
              1 hour quiz

7. Expectation

                        (a) Expectation of sums of r.v.s

                        (b) Moments

                        (c) Covariance, Variance of sums, Correlations

                        (d) Conditional expectation

                        (f) More one Normal Distribution

8. Limit laws

                        (a) Chebyshev’s inequality

                        (b) Strong law of large numbers, Weak law of large numbers

                        (c) The Central Limit Theorem