# week9 - Independence of random variables Definition Random...

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week 9 1 Independence of random variables Definition Random variables X and Y are independent if their joint distribution function factors into the product of their marginal distribution functions Theorem Suppose X and Y are jointly continuous random variables. X and Y are independent if and only if given any two densities for X and Y their product is the joint density for the pair ( X,Y ) i.e. Proof: If X and Y are independent random variables and Z =g ( X ), W = h ( Y ) then Z , W are also independent. ( ) ( ) ( ) y F x F y x F Y X Y X = , , ( ) ( ) ( ) y f x f y x f Y X Y X = , ,

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week 9 2 Example Suppose X and Y are discrete random variables whose values are the non- negative integers and their joint probability function is Are X and Y independent? What are their marginal distributions? Factorization is enough for independence, but we need to be careful of constant terms for factors to be marginal probability functions. () ... 2 , 1 , 0 , ! ! 1 , , = = + y x e y x y x p y x Y X μλ
week 9 3 Example and Important Comment The joint density for X , Y is given by •A r e X , Y independent?

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## This note was uploaded on 09/22/2010 for the course STATISTICS STA257 taught by Professor Mcdunnough during the Fall '10 term at University of Toronto.

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week9 - Independence of random variables Definition Random...

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