Lecture 12

Vector of Random Variables

Last Time (5/7) Pairs of R.Vs.

Functions of Two R.Vs

Expected Values

Conditional PDF

Reading Assignment: Sections 4.6-4.9

Probability & Stochastic ProcessesYates & Goodman (2nd Edition) NTUEE SCC_05_200812 - 1

Makeup Classes

I will attend Networking 2009 in Aachen, Germany, and need to make-up the classes of 5/14 & 5/15 (3 hours)

5/7 17:30 – 18:20, 5/8 8:10 – 9:00

5/21 17:30 – 18:20,

Probability & Stochastic ProcessesYates & Goodman (2nd Edition) NTUEE SCC_05_2008

12 - 2

Lecture 12: Random Vectors

Today (5/8)

Independence between Two R.Vs

Bivariate R.V.s Random Vector

Probability Models of N Random Variables

Vector Notation

Marginal Probability Functions

Independence of R.Vs and Random Vectors

Function of Random Vectors

Reading Assignment: Sections 4.10-5.5

Probability & Stochastic ProcessesYates & Goodman (2nd Edition) NTUEE SCC_05_2008

12 - 3

Lecture 12: Random Vector

Next Time: 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

Reading Assignment: Sections 5.5-6.3

Probability & Stochastic ProcessesYates & Goodman (2nd Edition) NTUEE SCC_04_2008

12 - 4

Objective

Analyze and monitor sensitivity of WAT parameter to In-

line then keep WAT unchanged by adjusting In-line shift.

Multiple Regression Model (MRM)

Correlation of Wafer Acceptance Test (WAT) and In-line

5

…..

Inline 1 Inline 2 Inline 3 Inline n WAT

model. theof residual is

3; 2, 1,for parameter, Inline is

3; 2, 1, 0,for parameter, estimated oft coefficien theis

parameter; WATis

3322110

e

jX

ib

Y

where

eXbXbXbbY

j

i

Manufacturing Process

Correlation Example

6

12 - 7

Brain Teaser: If You Were Kalman …

12 - 17

• Example: Let the number of men and women entering a post office in a certain interval be two independent Poisson random variables with parameters and , respectively. Find the conditional probability function of the number of men given the total number of persons.

Solution: Let N, M, K be the total number of men, women, and persons entering the post office. Note that K = M+N and M, N are independent. So we have K is also Poisson with parameter +.

pN|K(n|k) = P(N=n)P(M=k-n)/P(K=k)=

Probability and Stochastic ProcessesA Friendly Introduction for Electrical and Computer EngineersSECOND EDITION

Roy D. Yates David J. Goodman

Definitions, Theorems, Proofs, Examples, Quizzes, Problems, Solutions

Chapter 5

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