post2 - Change of Variables Bivariate case: Discrete: (X1,...

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Change of Variables Bivariate case: Discrete: ( X 1 ,X 2 ) f ( x 1 ,x 2 )on A y 1 = u 1 ( x 1 ,x 2 ) ,y 2 = u 2 ( x 1 ,x 2 ): 1-1 ( x 1 = w 1 ( y 1 ,y 2 ) ,x 2 = w 2 ( y 1 ,y 2 )) What is the pdf of ( Y 1 ,Y 2 )? g ( y 1 ,y 2 )= f [ w 1 ( y 1 ,y 2 ) ,w 2 ( y 1 ,y 2 )] , ( y 1 ,y 2 ) ∈B Continuous: ( X 1 ,X 2 ) h ( x 1 ,x 2 )on A y 1 = u 1 ( x 1 ,x 2 ) ,y 2 = u 2 ( x 1 ,x 2 ): 1-1 ( x 1 = w 1 ( y 1 ,y 2 ) ,x 2 = w 2 ( y 1 ,y 2 )) What is the joint pdf of ( Y 1 ,Y 2 )? 1 For ( y 1 ,y 2 ) ∈B , g ( y 1 ,y 2 )= h ( w 1 ( y 1 ,y 2 ) ,w 2 ( y 1 ,y 2 )) | J | , J = ± ± ± ± ± ± ± ± ± ∂x 1 ∂y 1 ∂x 1 ∂y 2 ∂x 2 ∂y 1 ∂x 2 ∂y 2 ± ± ± ± ± ± ± ± ± Example (p173) X 1 ,X 2 : a random sample from f ( x )= I (0 , 1) ( x ) Y 1 = X 1 + X 2 ,Y 2 = X 1 - X 2 Joint pdf of ( Y 1 ,Y 2 )? g ( y 1 ,y 2 )= 1 2 , ( y 1 ,y 2 ) ∈B B = { ( y 1 ,y 2 ): - 1 <y 2 < 1 , max( - y 2 ,y 2 ) < y 1 < min(2 + y 2 , 2 - y 2 ) } . 2
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post2 - Change of Variables Bivariate case: Discrete: (X1,...

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