LEC_EAS305_F10_1004

LEC_EAS305_F10_1004 - Independent Random Variables Moment...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Independent Random Variables Moment Generating Functions Bernoulli, Binomial, Hypergeometric Distrns Other Discrete Distributions Fall 2010 EAS 305 Lecture Notes Prof. Jun Zhuang University at Buffalo, State University of New York October 4, 6, 8, ... 2010 Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 1 of 68 Independent Random Variables Moment Generating Functions Bernoulli, Binomial, Hypergeometric Distrns Other Discrete Distributions Agenda for Today 1 Independent Random Variables Covariance and Correlation Theorems Examples Random Samples 2 Moment Generating Functions Introduction Other Applications 3 Bernoulli, Binomial, Hypergeometric Distrns Bernoulli( p ) Distribution Binomial( n , p ) Distribution Hypergeometric Distribution 4 Other Discrete Distributions Geometric and Negative Bionomial Distribution Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 2 of 68 Independent Random Variables Moment Generating Functions Bernoulli, Binomial, Hypergeometric Distrns Other Discrete Distributions Covariance and Correlation Theorems Examples Random Samples Introduction / Definition Recall that two events are independent if Pr( A B ) = Pr( A )Pr( B ). Then Pr( A | B ) = Pr( A B ) Pr( B ) = Pr( A )Pr( B ) Pr( B ) = Pr( A ) . And similarly, Pr( B | A ) = Pr( B ). Now want to define independence for RVs, i.e., the outcome of X doesnt influence the outcome of Y . Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 3 of 68 Independent Random Variables Moment Generating Functions Bernoulli, Binomial, Hypergeometric Distrns Other Discrete Distributions Covariance and Correlation Theorems Examples Random Samples Independent RV Definition: X and Y are independent RVs if, for all x and y , f ( x , y ) = f X ( x ) f Y ( y ) . Equivalent definitions: F ( x , y ) = F X ( x ) F Y ( y ) , x , y or Pr( X x , Y y ) = Pr( X x )Pr( Y y ) , x , y If X and Y are not indep, then theyre dependent . Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 4 of 68 Independent Random Variables Moment Generating Functions Bernoulli, Binomial, Hypergeometric Distrns Other Discrete Distributions Covariance and Correlation Theorems Examples Random Samples Theorem Theorem If X and Y are indep, then f ( y | x ) = f Y ( y ) . Proof: f ( y | x ) = f ( x , y ) f X ( x ) = f X ( x ) f Y ( y ) f X ( x ) = f Y ( y ) . Similarly, X and Y indep implies f ( x | y ) = f X ( x ). Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 5 of 68 Independent Random Variables Moment Generating Functions Bernoulli, Binomial, Hypergeometric Distrns Other Discrete Distributions Covariance and Correlation Theorems Examples Random Samples Example (discrete): f ( x , y ) = Pr( X = x , Y = y )....
View Full Document

This note was uploaded on 10/31/2010 for the course EE 423 taught by Professor Mitin during the Spring '10 term at SUNY Buffalo.

Page1 / 68

LEC_EAS305_F10_1004 - Independent Random Variables Moment...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online