Chapter 5 Section 5 6 and 7(1).pdf - 5.5 Independent Random...

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5.5 Independent Random Variables Recall that two events are independent if ( | ) ( ) P A B P A , by definition. Additionally, we have a theorem that states that two events are independent if ( ) ( ) ( ) P A B P A P B . We will use this second statement as our definition of independent random variables. Definition: Let X and Y be two random variables with joint pdf (or pmf) , ( , ) X Y f x y . The random variables X and Y are said to be independent if , ( , ) ( ) ( ) X Y X Y f x y f x f y . That is if the joint factors into the product of the marginals. Theorem: If X and Y are independent random variables, then , ( , ) ( ) ( ) X Y X Y F x y F x F y . Proof: , ( , ) ( , ) ( ) ( ) ( ) ( ) ( ) ( ) a b a b a b a X Y X Y X Y Y X F a b f x y dydx f x f y dydx f x f y dy dx F b f x dx            ( ) ( ) X Y F a F b . Example: The joint pmf of X and Y is given below. Earlier we had determined ( ) X f x and ( ) Y f y . If X and Y are independent, then , ( , ) ( ) ( ) X Y X Y f x y f x f x for all pairs ( , ) x y . We check the point (0,0) . , (0,0) .01 (.39)(.15) (0) (0) X Y X Y f f f , thus X and Y are not independent. Had the product been equal to the joint, we would still need to check the equality at all points in the support. Example: The joint pmf of the discrete random variables X, Y , and Z is given by 2 3 , , 1 ( , , ) ( ) 504 X Y Z f x y z x y z for 1,2,3 1,2,3 1,2,3 x y z
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