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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 ProcessesYates & Goodman (2nd Edition) NTUEE SCC_06_200914 - 1
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 ProcessesYates & Goodman (2nd Edition) NTUEE SCC_06_2009
13 - 2
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 ProcessesYates & Goodman (2nd Edition) NTUEE SCC_06_2009
14 - 3
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 ProcessesYates & Goodman (2nd Edition) NTUEE SCC_06_2009
14 - 4
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
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
if X(s) is defined for all values of s in some interval (-,
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.
Related Concepts
Probability Generating Function
X: D.R.V.
X N
Characteristic Function
1
( ) ( ) |z| 1iX
i
G z P X i z
Section 6.4
Sums of Independent R.Vs
Theorem 7.1
E[Mn(X)] = E[X]
Var[Mn(X)] = Var[X]/n
14 - 45
7.2 Deviation of a Random Variable from the Expected Value
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
Interpretation of Law of Large Numbers
14 - 54