# 09_19ans - STAT 410 Examples for Fall 2008 1 Let X and Y...

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STAT 410 Examples for 09/19/2008 Fall 2008 1. Let X and Y have the joint p.d.f. f X Y ( x , y ) = 20 x 2 y 3 , 0 < x < 1, 0 < y < x . a) Find f X ( x ), f Y ( y ). f X ( x ) = 5 x 4 , 0 < x < 1. f Y ( y ) = ( ) 9 3 3 20 y y - , 0 < y < 1. b) Find f X | Y ( x | y ), f Y | X ( y | x ). f X | Y ( x | y ) = 6 2 1 3 y x - , y 2 < x < 1. f Y | X ( y | x ) = 2 3 4 x y , 0 < y < x . c) Find E ( X | Y = y ), E ( Y | X = x ). E ( X | Y = y ) = 6 8 1 1 4 3 y y - - , 0 < y < 1. E ( Y | X = x ) = x 5 4 , 0 < x < 1. d) Find E ( X ), E ( Y ). E ( X ) = 6 5 . E ( Y ) = 11 8 .

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2. Let λ > 0. Consider the following joint probability distribution p ( x , y ) of two random variables X and Y: p ( x , y ) = ( ) ! 1 + - x e x , x , y – integers, 0 y x . a) Verify that p ( x , y ) is a legitimate probability mass function. 1. p ( x , y ) 0 for all ( x , y ). 2. ( ) + = = -
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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.

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09_19ans - STAT 410 Examples for Fall 2008 1 Let X and Y...

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