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Probability 2008

Probability 2008. Rong-Jaye Chen. Course Resources. Webpage: http://www.cs.nctu.edu.tw/~rjchen/Probability2008/. Text. S. Ghahramani, Fundamentals of Probability with stochastic processes 3rd Ed, Prentice-Hall, 2005. Grading Scheme. 1st Midterms 30%

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Probability 2008

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  1. Probability 2008 Rong-Jaye Chen

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

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

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

  5. 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/

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

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

  8. Probability 1 • Pr(atleast two with the same birthday among 23 people)=?

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

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

  12. 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. Sol: Let X1 and X2 be the points selected at random. Calculate f12(x, y), and use order statistic probilities in Sec 9.2 Sol = Pr(X(2)>=3X(1)) = … =1/3

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

  15. 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!

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