Lecture 14 Sums of Random Variables Last Time

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

Final Exam Announcement n Scope: Chapters 4 - 7 n 6/18 15: 30 – 17: 30 n HW#7 (no need to turn in) n 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 (2 nd Edition) 13 - 2 NTUEE SCC_06_2009

Lecture 14: Sums of R. V. s Today: n Sums of R. V. s n Moment Generating Functions n MGF of the Sum of Indep. R. Vs n Sample Mean (7. 1) n Deviation of R. V. from the Expected Value (7. 2) n Law of Large Numbers (part of 7. 3) n Central Limit Theorem Reading Assignment: Sections 6. 3 - 6. 6, 7. 1 -7. 3 Probability & Stochastic Processes Yates & Goodman (2 nd Edition) 14 - 3 NTUEE SCC_06_2009

Lecture 14: Sum of R. V. s Next Time: n Central Limit Theorem (Cont. ) n Application of the Central Limit Theorem n The Chernoff Bound n Point Estimates of Model Parameters n Confidence Intervals Reading Assignment: 6. 6 – 6. 8, 7. 3 -7. 4 Probability & Stochastic Processes Yates & Goodman (2 nd Edition) 14 - 4 NTUEE SCC_06_2009

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

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

if FX(s) is defined for all values of s in some interval (-d, d), d>0

Equal MGF same distribution Theorem Let X and Y be two random variables with moment-generating functions FX(s) and FY(s). If for some d > 0, FX(s) = FY(s) for all s, -d<s<d, then X and Y have the same distribution. n

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

Section 6. 4 Sums of Independent R. Vs

Theorem 7. 1 E[Mn(X)] = E[X] Var[Mn(X)] = Var[X]/n

7. 2 Deviation of a Random Variable from the Expected Value 14 - 45

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] n Bernoulli and Law of Large Number. pdf n n S&WLLN. doc

Interpretation of Law of Large Numbers 14 - 54