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

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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

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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

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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) = ?

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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?

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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

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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

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