Unformatted text preview: analogue of g ( y ).) Using the change of variables theorem from multivariable calculus, we have f Y 1 ,Y 2 ( y 1 ,y 2 ) = f X 1 ,X 2 ( x 1 ,x 2 )  J 1 . Example. Suppose X 1 is N (0 , 1), X 2 is N (0 , 4), and X 1 and X 2 are independent. Let Y 1 = 2 X 1 + X 2 ,Y 2 = X 13 X 2 . Then y 1 = g 1 ( x 1 ,x 2 ) = 2 x 1 + x 2 ,y 2 = g 2 ( x 1 ,x 2 ) = x 1x 3 , so J = ± 2 1 13 ² =7 . (In general, J might depend on x , and hence on y .) Some algebra leads to x 1 = 3 7 y 1 + 1 7 y 2 , x 2 = 1 7 y 12 7 y 2 . Since X 1 and X 2 are independent, f X 1 ,X 2 ( x 1 ,x 2 ) = f X 1 ( x 1 ) f X 2 ( x 2 ) = 1 √ 2 π ex 2 1 / 2 1 √ 8 π ex 2 2 / 8 . Therefore f Y 1 ,Y 2 ( y 1 ,y 2 ) = 1 √ 2 π e( 3 7 y 1 + 1 7 y 2 ) 2 / 2 1 √ 8 π e( 1 7 y 12 7 y 2 ) 2 / 8 1 7 . 22...
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This note was uploaded on 12/29/2011 for the course MATH 317 taught by Professor Wen during the Spring '09 term at SUNY Stony Brook.
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