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Unformatted text preview: STAT 410 Examples for 09/12/2008 Fall 2008 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 0.15 0 2 0.15 0.35 0.20 a) Find P ( X + Y = 2 ). P ( X + Y = 2 ) = p ( 1, 1 ) + p ( 2, 0 ) = 0.15 + 0.15 = 0.30 . b) Find P ( X > Y ). P ( X > Y ) = p ( 1, 0 ) + p ( 2, 0 ) + p ( 2, 1 ) = 0.15 + 0.15 + 0.35 = 0.65 . 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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This note was uploaded on 10/22/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Fall '08 term at University of Illinois at Urbana–Champaign.
- Fall '08