chapter5_1 - Section 5.1: Joint Distribution of Random...

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Section 5.1: Joint Distribution of Random Variables (Only Discrete Case) 1
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Concepts and Formulae: Let X and Y be discrete random variables. The joint PMF of X and Y is defined by p ( x,y ) = P ( X = x,Y = y ) . Let A be any set consisting of pairs ( ) val- ues. Then, P [( X,Y ) A ] = XX ( x,y ) A p ( ) . The marginal PMF of X is p X ( x ) = X y p ( ) and the marginal PMF of Y is p Y ( y ) = X x p ( ) . X and Y are independent if for all ( ) p ( ) = p X ( x ) p Y ( y ) . The conditional PMF of X given Y is p X | Y ( x | y ) = p ( ) p Y ( y ) and the conditional PMF of Y given X is p Y | X ( y | x ) = p ( ) p X ( x ) . 2
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First example of Section 5.1. Examples 5.1 and 5.2 on textbook. The (joint) PMF of X and Y is y x 0 100 200 Total 100 0 . 2 0 . 1 0 . 2 0 . 5 250 0 . 05 0 . 15 0 . 30 0 . 5 0 . 25 0 . 25 0 . 5 1 P ( Y 100) =0 . 1 + 0 . 2 + 0 . 15 + 0 . 30 =0 . 75 . P ( X + Y 300) =0 . 2 + 0 . 1 + 0 . 2 + 0 . 05 =0 . 55 . The marginal PMF of X is x 100 250 p X ( x ) 0 . 5 0 . 5 and the marginal PMF of Y is y 0 100 200 p Y ( y ) 0 . 25 0 . 25 0 . 5 3
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Are X and Y independent? Answer: since p (100 , 0) = 0 . 2 6 = p X (100) p Y (0) = 0 . 5 × 0 . 25 X and Y are not independent.
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chapter5_1 - Section 5.1: Joint Distribution of Random...

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