jointnotes

# jointnotes - Joint Probability Distributions...

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Joint Probability Distributions – De…nitions/Notes De…nition 1 Let X and Y be two discrete random variables de…ned on the sample space S of an experiment. The joint probability mass function p ( x;y ) is de…ned for each pair of numbers ( x;y ) by: p ( x;y ) = P ( X = x and Y = y ) : Let A be any set consisting of pairs of ( x;y ) values. Then the probability P [( X;Y ) 2 A ] is obtained by summing the joint pmf over pairs in A : P [( X;Y ) 2 A ] = XX ( x;y ) 2 A p ( x;y ) : De…nition 2 The marginal probability mass functions of X and Y , denoted by p X ( x ) and p Y ( y ) , respectively, are given by: p X ( x ) = X y p ( x;y ) and: p Y ( y ) = X x p ( x;y ) : De…nition 3 Let X and Y be continuous random variables. Then f ( x;y ) is the joint probability density function for X and Y if for any two-dimensional set A : P [( X;Y ) 2 A ] = Z Z A f ( x;y ) dx dy: In particular, if A is the two-dimensional rectangle f ( x;y ) : a · x · b;c · y · d g , then: P [( X;Y ) 2 A ] = P

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jointnotes - Joint Probability Distributions...

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