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08_31 - STAT 410 Examples for Fall 2011 Multivariate...

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STAT 410 Examples for 08/31/2011 Fall 2011 Multivariate Distributions Let X and Y be two discrete random variables. The joint probability mass function p ( x , y ) is defined 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 P ( ( X, Y ) A ) = ( ) ( ) ∑ ∑ y x A y x p , , . Let X and Y be two 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 ) A ) = ( ) ∫∫ A dy dx y x f , . 1. Consider the following joint probability distribution p ( x , y ) of two random variables X and Y: x \ y 0 1 2 1 0.15 0.10 0 2 0.25 0.30 0.20 a) Find P ( X + Y = 2 ) . b) Find P ( X > Y ) . The marginal probability mass functions of X and of Y are given by p X ( x ) = ( ) y y x p all , , p Y ( y ) = ( ) x y x p all , . The marginal probability density functions of X and of Y are given by f X ( x ) = ( ) - , dy y x f , f Y ( y ) = ( ) - , dx y x f .
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c) Find the (marginal) probability distributions p X ( x ) of X and p Y ( y ) of Y.
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