# 10_06_1ans - STAT 400 Examples for 10/06/2011 (1) Fall 2011...

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STAT 400 Examples for 10/06/2011 (1) 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 ). P ( X > Y ) = p ( 1, 0 ) + p ( 2, 0 ) + p ( 2, 1 ) = 0.15 + 0.25 + 0.30 = 0.70 . b) Find P ( X + Y = 2 ). P ( X + Y = 2 ) = p ( 1, 1 ) + p ( 2, 0 ) = 0.10 + 0.25 = 0.35 .

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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 . c) Find the (marginal) probability distributions p X ( x ) of X and p Y ( y ) of Y. y p Y ( y ) x p X ( x ) 0 0.40 1 0.25 1 0.40 2 0.75 2 0.2 If p ( x , y ) is the joint probability mass function of ( X, Y ) OR f ( x , y ) is the joint probability density function of ( X, Y ), then discrete continuous E ( g ( X, Y ) ) = ∑ ∑ x y y x p y x g all all ) , ( ) , ( E ( g ( X, Y ) ) = ∫ ∫ - - dy dx y x f y x g ) , ( ) , ( d) Find E ( X ), E ( Y ), E ( X + Y ), E ( X Y ). E
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## This note was uploaded on 11/07/2011 for the course STAT 400 taught by Professor Kim during the Fall '08 term at University of Illinois, Urbana Champaign.

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10_06_1ans - STAT 400 Examples for 10/06/2011 (1) Fall 2011...

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