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Jointly distributed random variables Joint distributions and the central limit theorem Sayan Mukherjee Sta. 113 Chapter 5 of Devore September 27, 2007 Sayan Mukherjee Joint distributions and the central limit theorem
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Jointly distributed random variables Table of contents 1 Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Sayan Mukherjee Joint distributions and the central limit theorem
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Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Joint distributions Definition Let X , Y be two discrete random variables. The joint pdf p ( x , y ) is defined by p ( x , y ) = IP ( X = x and y = Y ) , and for a set A IP [( x , y ) A ] = X x , y A p ( x , y ) . Sayan Mukherjee Joint distributions and the central limit theorem
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Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Example An evil Leprechaun is chopping of fingers and toes at night. People wake up with X = 1 , 2 fingers and Y = 2 , 3 , 4 toes. Sayan Mukherjee Joint distributions and the central limit theorem
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Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Example p ( x , y ) y = 2 y = 3 y = 4 x = 1 . 2 . 1 . 2 x = 2 . 05 . 15 . 3 IP ( y > 2) = p ( x = 1 , y = 3) + p ( x = 1 , y = 4) + p ( x = 2 , y = 3) + p ( x = 2 , y = 4) = . 1 + . 2 + . 15 + . 3 = . 75 IP ( x = 2 , y < 4) = p ( x = 2 , y = 3) + p ( x = 2 , y = 2) = . 15 + . 05 = . 2 Sayan Mukherjee Joint distributions and the central limit theorem
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Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Marginal distributions Definition The marginal distributions of p ( x , y ) denoted by p X ( x ) and p Y ( y ) are given by p X ( x ) = X y p ( x , y ) p Y ( y ) = X x p ( x , y ) . Sayan Mukherjee Joint distributions and the central limit theorem
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Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Example p ( x , y ) y = 2 y = 3 y = 4 x = 1 . 2 . 1 . 2 x = 2 . 05 . 15 . 3 Sayan Mukherjee Joint distributions and the central limit theorem
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Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Example p ( x , y ) y = 2 y = 3 y = 4 p X ( x ) x = 1 . 2 . 1 . 2 .5 x = 2 . 05 . 15 . 3 .5 Sayan Mukherjee Joint distributions and the central limit theorem
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Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Example p ( x , y ) y = 2 y = 3 y = 4 x = 1 . 2 . 1 . 2 x = 2 . 05 . 15 . 3 p Y ( y ) . 25 . 25 . 5 Sayan Mukherjee Joint distributions and the central limit theorem
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Jointly distributed random variables Discrete random variables Continuous random variables Covariance A statistic Central limit theorem Joint distributions Definition Let X , Y be two continuous random variables. The joint pdf p ( x , y ) as the function that for any set A IP [( x , y ) A ] = ZZ A p ( x , y ) dx dy .
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