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# chapter5 - Chapter 5 Joint Probability Distribution and...

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Chapter 5 Joint Probability Distribution and Random Sample STA 113 Spring 2008 Jayanta Kumar Pal 1 / 28

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Chapter 5 Outline 1 Introduction 2 Expected values, Covariance and Correlation 3 Statistics and their distributions 2 / 28
Chapter 5 Introduction Outline 1 Introduction 2 Expected values, Covariance and Correlation 3 Statistics and their distributions 3 / 28

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Chapter 5 Introduction Beyond single random variable. Instead of p ( x ) or f ( x ) , we move to p ( x , y ) or f ( x , y ) and beyond. For discrete RV : joint pmf. For continuous RV : joint pdf. Suppose ( X , Y ) are both discrete. We know p ( x ) , p ( y ) completely. But, we cannot compute P ( X Y ) or P ( 5 X , Y = 0 ) , the probability of any event that involves both the RV. 4 / 28
Chapter 5 Introduction Joint Probability Mass Function S - sample space. ( X , Y ) - discrete RV deﬁned on S . Joint pmf of ( X , Y ) , deﬁned for each pair of possible values ( x , y ) p ( x , y ) = P ( X = x , Y = y ) p ( x , y ) 0, x y p ( x , y ) = 1. For any A , set of paired values ( x , y ) , P [( X , Y ) A ] = X X ( x , y ) A p ( x , y ) e.g. let both X , Y take values 0 , 1 , 2 ,... and p ( x , y ) be the joint pmf deﬁned on D = { ( x , y ) : x , y = 0 , 1 , 2 ,... } . A = { ( x , y ) : x + y = 2 } = { ( 0 , 2 ) , ( 1 , 1 ) , ( 2 , 0 ) } . Then, P [( X , Y ) A ] = P ( X + Y = 2 ) = p ( 0 , 2 ) + p ( 1 , 1 ) + p ( 2 , 0 ) . 5 / 28

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Chapter 5 Introduction Joint probability table For a manageable and ﬁnite number of possible ( x , y ) values, p ( x , y ) is usually represented in a table. Rows correspond to different values of X , columns for Y . The cell entries : p ( x , y ) . Row totals : y p ( x , y ) = p X ( x ) Column totals : x p ( x , y ) = p Y ( y ) Marginal pmf’s for X and Y respectively. Used to compute probabilities related to single variables, e.g. P ( X 3 ) = 3 x = 0 p X ( x ) etc x p X ( x ) = y p Y ( y ) = 1. 6 / 28
Chapter 5 Introduction Continuous Random Variables Joint pdf of two RV’s : f ( x , y ) deﬁned for all pairs ( x , y ) . f ( x , y ) is the joint pdf of a continuous pair ( X , Y ) if for any set A R 2 , P [( X , Y ) A ] = Z Z ( x , y ) A f ( x , y ) dx dy f ( x , y ) 0, R -∞ R -∞ f ( x , y ) dx dy = 1. For a rectangle A = { ( x , y ) : a x b , c y d } , P [( X , Y ) A ] = P ( a X b , c Y d ) = Z b a Z d c f ( x , y ) dy dx 7 / 28

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Chapter 5 Introduction Marginal density functions The marginal pdf of X and Y are f X ( x ) = Z -∞ f ( x , y ) dy , and f Y ( y ) = Z -∞ f ( x , y ) dx respectively. R
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chapter5 - Chapter 5 Joint Probability Distribution and...

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