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LEC_EAS305_F10_1004

# LEC_EAS305_F10_1004 - Independent Random Variables Moment...

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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

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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 RV’s, i.e., the outcome of X doesn’t influence the outcome of Y . Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 3 of 68

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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 RV’s 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 they’re 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

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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 ). X = 1 X = 2 f Y ( y ) Y = 2 .12 .28 .4 Y = 3 .18 .42 .6 f X ( x ) .3 .7 1 X and Y are indep since f ( x , y ) = f X ( x ) f Y ( y ), x , y . Prof. Jun Zhuang Fall 2010 EAS 305 Lecture Notes Page 6 of 68
Independent Random Variables Moment Generating Functions Bernoulli, Binomial, Hypergeometric Distrns Other Discrete Distributions Covariance and Correlation Theorems Examples Random Samples Example (cts): Suppose f ( x , y ) = 6 xy 2 , 0 x 1, 0 y 1.

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• Spring '10
• MITIN
• Probability theory, Binomial distribution, Discrete probability distribution, Geometric distribution, Prof. Jun Zhuang

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