# mult - Multivariate(actually mostly bivariate Distributions...

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Multivariate (actually mostly bivariate!) Distributions DISCRETE: Let X and Y be discrete random variables. Then the joint pmf of X and Y is p X,Y ( x,y ) = P ( X = x,Y = y ) . [Note: Read P ( X = x,Y = y ) as P ( X = x and Y = y ).] An important property that all joint pmf’s should satisfy is : X x X y p X,Y ( x,y ) = 1 . The marginal pmf of X is p X ( x ) = X y p X,Y ( x,y ) = P ( X = x ) and the marginal pmf of Y is p Y ( y ) = X x p X,Y ( x,y ) = P ( Y = y ) . The following are the respective conditional pmf’s p X | Y ( x | y ) = P ( X = x | Y = y ) = P ( X = x,Y = y ) P ( Y = y ) = p X,Y ( x,y ) p Y ( y ) p Y | X ( y | x ) = P ( Y = y | X = x ) = P ( X = x,Y = y ) P ( X = x ) = p X,Y ( x,y ) p X ( x ) Let k ( X,Y ) be a function of the random variables, then the expected value (or mean or average value) of k ( X,Y ) is E [ k ( X,Y )] = X x X y k ( x,y ) p X,Y ( x,y ) . The covariance of the random variables X and Y , often written as Cov ( X,Y ) or σ XY , and is given by Cov ( X,Y ) = E [( X - E ( X ))( Y - E ( Y ))] = E [ XY ] - E [ X ] E [ Y ] . If

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## This note was uploaded on 09/18/2011 for the course ISEN 609 taught by Professor Klutke during the Spring '08 term at Texas A&M.

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mult - Multivariate(actually mostly bivariate Distributions...

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