1. Lecture 14 Sums of Random Variables Last Time (5/21, 22) Pairs Random Vectors Function of Random Vectors Expected Value Vector and Correlation Matrix Gaussian Random Vectors Sums of R. V.s Expected Values of Sums PDF of the Sum of Two R.V.s Moment Generating Functions Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_06_2009 14 - 1

2. Final Exam Announcement Scope: Chapters 4 - 7 6/18 15:30 – 17:30 HW#7 (no need to turn in) Problems of Chapter 7 7.1.2, 7.1.3, 7.2.2, 7.2.4, 7.3.1, 7.3.4, 7.3.6 7.4.1, 7.4.3, 7.4.4, 7.4.6 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_06_2009 13 - 2

3. Lecture 14: Sums of R.V.s Today: Sums of R. V.s Moment Generating Functions MGF of the Sum of Indep. R.Vs Sample Mean (7.1) Deviation of R. V. from the Expected Value (7.2) Law of Large Numbers (part of 7.3) Central Limit Theorem Reading Assignment: Sections 6.3- 6.6, 7.1-7.3 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_06_2009 14 - 3

4. Lecture 14: Sum of R.V.s Next Time: Central Limit Theorem (Cont.) Application of the Central Limit Theorem The Chernoff Bound Point Estimates of Model Parameters Confidence Intervals Reading Assignment: 6.6 – 6.8, 7.3-7.4 Probability & Stochastic Processes Yates & Goodman (2nd Edition) NTUEE SCC_06_2009 14 - 4

5. Brain Teaser 1: Stock Price Trend Analysis Stock price variation per day: P(rise) = p, P(fall)=1-p If rise, the percentage is exp~ Prob(consecutive rise in n days and total percentage higher than x) = ? 14 - 5

6. Brain Teaser 2: Is Wang’s Stuff Back? Wang’s Stuff: the Sinker balls Speed Drop Wang said he is ready. If you were Giradi or Cashman, how do you know if he is ready? 15 - 6

8. if  X (s) is defined for all values of s in some interval (- , 

12. Equal MGF  same distribution Theorem Let X and Y be two random variables with moment-generating functions  X (s) and  Y (s). If for some  > 0,  X (s) =  Y (s) for all s, -  <s< , then X and Y have the same distribution.

13. Related Concepts Probability Generating Function X: D.R.V. X N Characteristic Function

17. Section 6.4 Sums of Independent R.Vs

43. Theorem 7.1 E[M n (X)] = E[X] Var[M n (X)] = Var[X]/n

45. 14 - 45 7.2 Deviation of a Random Variable from the Expected Value

53. Law of Large Numbers: Strong and Weak Jakob Bernoulli, Swiss Mathematician, 1654-1705 [Ars Conjectandi, Basileae, Impensis Thurnisiorum, Fratrum, 1713 The Art of Conjecturing; Part Four showing The Use and Application of the Previous Doctrine to Civil, Moral and Economic Affairs Translated into English by Oscar Sheynin, Berlin 2005] Bernoulli and Law of Large Number.pdf S&WLLN.doc

54. Interpretation of Law of Large Numbers 14 - 54