1. Probability 2008 Rong-Jaye Chen

2. p2. Course Resources Webpage: http://www.cs.nctu.edu.tw/~rjchen/Probability2008/

3. p3. Text S. Ghahramani, Fundamentals of Probability with stochastic processes 3rd Ed, Prentice-Hall, 2005

4. p4. Grading Scheme 1st Midterms 30% 2nd Midterms 30% Final Exam 40%

5. p5. TAs 1. TAs 徐順隆 曾輔國 2. 霹靂博課程資訊 : 霹靂博 : 蔡佩娟 上課時間 -4HY, 上課地點 -EC114, Office hour-2EF@EC119 網頁 : http://cube.csie.nctu.edu.tw/~pctsai/prob2008/ http://cube.csie.nctu.edu.tw/~pctsai/prob2008/

6. p6. Syllabus Axioms of probability Combinatorial methods Conditional probability and independence Distributed functions and discrete random variables Special discrete distributions

7. p7. Continuous random variables Special continuous distributions Bivariate distribution Multivariate distribution More expectations and variances Sums of independent random variables and limit theorems

8. p8. Probability 1 Pr(at least two with the same birthday among 23 people)=?

9. p9. Sol: x =1-Pr(each two among 23 with different birthdays) =1-(365/365)(364/365) … (343/365) > 0.5 (Surprised?) It is called Birthday Paradox!

10. p10. Probability 2 On average, there are three misprints in every 10 pages of a particular book. If Chapter 1 contains 35 pages, what is the probability that Chapter 1 has 10 misprints?

11. p11. Sol: lamda=(3/10)35=10.5 It is a Poisson distribution so solution = e -10.5 (10.5) 10 /10!=0.124

12. p12. Probability 3 Two random points are selected from (0,1) independently. Find the probability that one of them is at least three times the other.

13. p13. Sol: Let X 1 and X 2 be the points selected at random. Calculate f 12 (x, y), and use order statistic probilities in Sec 9.2 Sol = Pr(X (2) >=3X (1) ) = … = 1/3

14. p14. Probability 4 Toss a coin 10000 times, #(head)=5150 times. Is the coin unbiased?

15. p15. Sol: Suppose this coin is unbiased X=#(head) ~Binomial(n=10000,p=0.5) E(X)=np=5000 Var(X)=npq=2500 By Central Limit Theorem Pr(X>=5150) ~ Pr(Z>3)~0.001 (Z is the normal distributed random variable) So reject the hypothesis! The coin is biased!