1 / 54

Lecture 14 Sums of Random Variables

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.

pcampbell
Download Presentation

Lecture 14 Sums of Random Variables

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 14Sums 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

  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

  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

  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

  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~l • Prob(consecutive rise in n days and total percentage higher than x) = ?

  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?

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

  8. 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.

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

  10. Section 6.4 Sums of Independent R.Vs

  11. Theorem 7.1 E[Mn(X)] = E[X] Var[Mn(X)] = Var[X]/n

  12. 7.2 Deviation of a Random Variable from the Expected Value 14 - 45

More Related