（英文）：Introduction to Probability

 學年： 九十七學年 第一學期 授課班 級： 數學系三 全/半年： 半 必/選修：必 學分： 3 上課時間：星期四下午 2:10 至 5:00 上課地點： 教師聯絡方式：數學館 M205 Office Hours: 週三 9-11 教學網頁：http://math.ntnu.edu.tw/~pwtsai/prob97/index.html E-mail: pwtsai@math.ntnu.edu.tw 助教姓名： 助教 Office Hours： 助教聯絡方式：數學館 M 410

### 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：

Hogg, R.V. & Tanis, E. A. (2006) Probability and Statistical Inference, 7th edition, Prentice Hall.
Ross, S. (2006) A First Course in Probability 7th ed. Pearson Prentice Hall.

Syllabus
 HT 1. Combinatorial Analysis (a) Counting techniques: multiplication principle, permutations, combinations (b) Multinomial Coefﬁcients 1.3 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. 1.1 1.2 3. Conditional probability (a) Conditional probability (b) Bayes’ Formula (c) Independent events 1.4 1.5 1.6 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, Uniform Distribution, Geometric Distribution, Negative Binomial Distribution, Hypergeometric Distribution, Poisson Distribution, (f) Poisson approximation to the Binomial (g) The moment generating function ch2 Midterm exam 5.Continuous Distribution (a) Continuous-type data (b) Some continuous distribution: Uniform Distribution, Exponential Distribution, Normal Distribution, Cauchy Distribution (c) Gamma and Chi-square Distribution (d) Normal approximation to the binomial ch3  5.1 5.2 5.5 quiz 1 (1 hr) 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 (f) Distributions of functions of a random variable: Normal to Chi-square Distribution, Lognormal 4.1 4.2 4.3 4.4 4.5 quiz 2 (1 hr) 7. Expectation (a) Expectation of sums of r.v.s (b) Covariance, Variance of sums, Correlations (c) Conditional expectation (d) More one Normal Distribution 4.6 5.3 5.6 8. Limit laws (a) Chebyshev’s inequality (b) Strong law of large numbers, Weak law of large numbers (c) The Central Limit Theorem 4.7 5.4 Final exam

### 三、實施 方式：

課 堂講授
 Lecture notes HW 9/18 Ch1.pdf 9/25 hw97_1.pdf (due 10/2) 9/25修正後 10/2 Ch2.pdf hw97_2.pdf (due 10/9) 10/9 2.1-3(a,d), , 2.1-11., 2.2-2, 2.2-10 2.3-13 2.3-14 10/16 2.4-3, 2.4-6, 2.4-10 2.4-23 2.5-2 (a) (d), 2.5-4, 2.5-8 2.5-10, 2.5-11, 2.5-21 10/23 HW: due 2-1 ~ 2-5 2.6-3, 2.6-4, 2.6-8, 2.6-10 10/30 期中考 mid1.pdf 11/6 Ch3.pdf some graphs 11/13 3.2-3.3 and 3.4 gamma 3.2-1(c), 3.2-6, 3.2-8 3.3-3, 3.3-4, 3.3-5, 3.3-10, 3.4-7, 3.4-8, 3.4-12 11/20 3.4 Chi-square 3.5, 5.2 11/27 小考 3.3, 3.4, 3.5, 5.2 Ch401.pdf 3.4-11, 3.4-16, 5.2-11, 5.2-18(b) 3.5-2, 3.5-10, 3.5-11 5.2-5(b), 5.2-6(c), 5.2-10, 5.2-24, 5.2-26 12/4 Ch402.pdf 12/11 Ch403.pdf 4.1-1, 4.1-9, 4.1-12 4.2-3(a),(b), 4.2-7, 4.2-8(b), (c), (d). 4.2-12 (a) (b) 4.3-1(e) 4.4-13(d) 4.3-16(c)  (due 12/11) 12/18 4.4-1, 4.4-8, 4.4-14 4.5-2, 4.5-3, 4.5-6, 4.5-7, 4.5-9, 4.5-15. (due 12/30) 12/25 小考 4.1-4.4 請看講義範圍 ch401.pdf & ch402.pdf quiz2_sol.pdf 1/1 4.6-1, 4.6-6, 4.6-26 4.7-1(a), 4.7-6, 5.2-17, 5.3-3, 5.3-5, 5.3-6 (due 1/8) 1/8 Ch5.pdf 5.3-8(b) 5.3-14, 5.3-18 5.4-10, 5.4-14. 5.5-1, 5.5-8, 5.5-11, 5.5-22 (no need to hand in) 1/15 期末考

### 四、評量 標準與成績計算方式：

(A) 兩 次考試 各 30%, 30% 及 兩 次小 考15%, 作業及平時成績25%
(B) 兩 次考試 各 40% 40%