# EX 9 - STAT 400 Spring 2011 Examples for 03/07/2011 Let X...

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STAT 400 Examples for 03/07/2011 Spring 2011 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. Alexis Nuts, Inc. markets cans of deluxe mixed nuts containing almonds, cashews, and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let X = the weight of almonds in a selected can and Y = the weight of cashews. Then the region of positive density is D = { ( x , y ) : 0 x 1, 0 y 1, x + y 1 } . Let the joint probability density function for ( X , Y ) be ( ) + = otherwise 0 1 , 1 0 , 1 0 60 , 2 y x y x y x y x f a) Verify that ( ) y x f , is a legitimate probability density function. 1.

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## This note was uploaded on 11/28/2011 for the course STAT 400 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.

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EX 9 - STAT 400 Spring 2011 Examples for 03/07/2011 Let X...

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