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# slide5 - ECON321 Econometrics Lecture 5 Joint Probability...

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ECON321 : Econometrics Lecture 5 : Joint Probability Distributions Sasan Bakhtiari University of Maryland, College Park Summer 2007 Session II,

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Two Discrete Random Variables I The joint probability distribution of two variables X and Y is p ( x , y ) = P ( X = x , Y = y ) I p ( x , y ) 0 , x , y , x y p ( x , y ) = 1. I Why joint distribution? Many variables are inter-related and cannot be assumed independent.
Example 5.1 A person’s height and weight are related. Taller people are normally heavier. Assume that the following Table lists the joint distribution of weight and height H 5 6 7 100 0.2 0.1 0.02 W 140 0.06 0.2 0.18 180 0.04 0.1 0.1 I What is P ( W = 180 , H = 5)? 0.04 I What is P ( W = 180 , H < 7)? P ( W = 180 , H < 7) = P ( W = 180 , H = 5) + P ( W = 180 , H = 6) = 0 . 04 + 0 . 1 = 0 . 14 .

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Marginal Distributions I Probability distributions of p ( x ) and p ( y ) are called the marginal distributions of p ( x , y ). I p ( x ) = y p ( x , y ) and p ( y ) = x p ( x , y ). I In the previous example p ( W = 100) = 0 . 2 + 0 . 1 + 0 . 02 = 0 . 32 p ( W = 140) = 0 . 06 + 0 . 2 + 0 . 18 = 0 . 44 p ( W = 180) = 0 . 04 + 0 . 1 + 0 . 1 = 0 . 24 I and p ( H = 5) = 0 . 2 + 0 . 06 + 0 . 04 = 0 . 3 p ( H = 6) = 0 . 1 + 0 . 2 + 0 . 1 = 0 . 4 p ( H = 7) = 0 . 02 + 0 . 18 + 0 . 1 = 0 . 3 I Do the above distributions satisfy the required axioms?
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