09_14ans - STAT 410 Examples for 09/14/2009 Fall 2009 2.5...

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STAT 410 Examples for 09/14/2009 Fall 2009 2.5 Independent Random Variables 1. Consider the following joint probability distribution p ( x , y ) of two random variables X and Y: y x 0 1 2 p X ( x ) 1 0.15 0.15 0 0.30 2 0.15 0.35 0.20 0.70 p Y ( y ) 0.30 0.50 0.20 1.00 a) Are events {X = 1} and {Y = 1} independent? P ( X = 1 Y = 1 ) = p ( 1, 1 ) = 0.15 = 0.50 × 0.30 = P ( X = 1 ) × P ( Y = 1 ). {X = 1} and {Y = 1} are independent . Def Random variables X and Y are independent if and only if discrete p ( x , y ) = p X ( x ) p Y ( y ) for all x , y . continuous f ( x , y ) = f X ( x ) f Y ( y ) for all x , y . b) Are random variables X and Y independent? p ( 1, 0 ) = 0.15 0.30 × 0.30 = p X ( 1 ) × p Y ( 0 ). X and Y are NOT independent .
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2. Let the joint probability density function for ( X , Y ) be ( 29 < + < < < < = otherwise 0 1 , 1 0 , 1 0 24 , y x y x y x y x f Recall: f X ( x ) = ( 29 2 1 12 x x - , 0 < x < 1, f Y ( y ) = ( 29 2 1 12 y y - , 0 < y < 1. Are random variables X and Y independent?
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09_14ans - STAT 410 Examples for 09/14/2009 Fall 2009 2.5...

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