06_03 - Proof. Let Ai (y) = cfw_x : gi (x) y, i=1, ., n,...

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p. 6-21 Proof . Let i =1, …, n , then A i ( y )= { x : g i ( x ) y } , F Y ( y 1 ,...,y n P ( Y 1 y 1 ,...,Y n y n ) = P ( X 1 A 1 ( y 1 ) ,...,X n A n ( y n )) = P ( X 1 A 1 ( y 1 )) ×···× P ( X n A n ( y n )) = P ( Y 1 y 1 ) P ( Y n y n ) = F Y 1 ( y 1 ) F Y n ( y n ) . • Theorem. X =( X 1 , …, X n ) are independent if and only if there exist univariate functions g i ( x ), i =1, …, n , such that (a) when X 1 , …, X n are discrete r.v.’s with joint pmf p X , p X ( x 1 , …, x n ) g 1 ( x 1 ) × × g n ( x n ), < x i < , i =1,…, n . (b) when X 1 , …, X n are continuous r.v.’s with joint pdf f X , f X ( x 1 , …, x n ) g 1 ( x 1 ) × × g n ( x n ), < x i < , i =1,…, n . p. 6-22 Example. If the joint pdf of ( X , Y ) is and f ( x , y )=0, otherwise, i.e., then X and Y are independent. Note that the region in which the joint pdf is nonzero can be expressed in the form {( x , y ): x A , y B }. f ( x, y ) e 2 x e 3 y , 0 <x< , 0 <y< , X Y
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p. 6-23 Reading : textbook, Sec 6.2 Suppose that the joint pdf of ( X , Y ) is and f ( x , y )=0, otherwise, i.e., X and Y are not independent. f ( x, y ) xy, 0 <x< 1 , 0 <y< 1 , 0 <x + y< 1 , X Y Q : For independent X and Y , how should their joint pdf/pmf look like? X Y X Y D Q : Given the joint distribution of X =( X 1 , …, X n ), how to find the distribution Y =( Y 1 , …, Y k ), where Y 1 = g 1 ( X 1 , …, X n ), …, Y k = g k ( X 1 , …, X n ), denoted by Y = g ( X ), g : R n R k .
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This note was uploaded on 02/15/2012 for the course MATH 2810 taught by Professor Shao-weicheng during the Fall '11 term at National Tsing Hua University, China.

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06_03 - Proof. Let Ai (y) = cfw_x : gi (x) y, i=1, ., n,...

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