10_06_2ans - STAT 400 4.1(continued 1 Fall 2011 Examples...

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Examples for 10/06/2011 (2) Fall 2011 4.1 (continued) Independent Random Variables 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 0.25 2 0.25 0.30 0.20 0.75 0.40 0.40 0.20 Recall: A and B are independent if and only if P ( A B ) = P ( A ) P ( B ). a) Are events {X = 1} and {Y = 1} independent? P ( X = 1 Y = 1 ) = p ( 1, 1 ) = 0.10 = 0.25 × 0.40 = 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 . F ( x , y ) = P ( X x , Y y ). f ( x , y ) = 2 F ( x , y ) / x y . Def Random variables X and Y are independent if and only if F ( x , y ) = F X ( x ) F Y ( y ) for all x , y . b)
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10_06_2ans - STAT 400 4.1(continued 1 Fall 2011 Examples...

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