（英文）：Introduction to Probability

 學年： 100學年 第二學期 授課班 級： 數學系二 全/半年： 半 必/選修：必 學分： 3 上課時間：星期五下午2:00 至 5:00 上課地點： C003 教師聯絡方式：數學館 M205 Office Hours: 週五 9-11 教學網頁：http://math.ntnu.edu.tw/~pwtsai/prob00/index.html E-mail: pwtsai@math.ntnu.edu.tw 助教: 洪辰諭：jerry111076@gmail.com 蕭煜霖：understar17@hotmail.com.com 助教 Office Hours：星期二及三中午

## 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);

 Lecture Notes Problems Theoretical Exercises Combinatorial Analysis prob00_ch1.pdf 7, 26, 28, 31 10,13 9,10,16 Probability prob00_ch2.pdf 2,3,4,8,15,27,37,45, 47,48,50 7, 11, 12, 13 14. Conditional probability and Independence prob00_ch3.pdf 1,4,14,26,29,47,56, 59,87 6,11 5,23 ch1-3_HW_sol.pdf Discrete Random Variables prob00_ch4.pdf 5, 8,17,18,21,25, 28, 37,38,50,55, 59,69, 75, 79 1,3,7,20 1,21 ch4_sol.pdf Continuous Random Variables prob00_ch5a.pdf 2,4,11, 7 4 ch5_sol_1.pdf 4/13 期中考 prob00_ch5.pdf 8,17, 23,32, 36, 40 13, 29,30 ch5_sol.pdf prob00_ch6.pdf 2,7,10,12,13,17, 20,27,29,38,39, 41,48,52,54,56 Buffon's needle, 9 16 ch6_sol.pdf prob00_ch7.pdf 4,11,16,33,38,39, 50, 71, 72, 75, 76 22,48, 51,55 ch7_sol.pdf ch7_sol_2.pdf prob00_ch8.pdf 2,4,7,9,13, 14 ch8_sol.pdf

### 考古題：

98mid1 9801  9802   9802mid   98final

### 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

Final exam

Pitman, J. (1999) Probability, Springer.

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