11 Example Class 9

11 Example Class 9 - THE UNIVERSITY OF HONG KONG DEPARTMENT...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 9 Review Transformation of Multivariate Distribution Let X 1 ,X 2 ,...,X n be jointly distributed continuous random variables with joint probability density function f x ( x 1 ,x 2 ,...,x n ). Also, let Y i = g i ( X 1 ,...,X n ) ,i = 1 , 2 ,...,n for some function g ’s which satisfy the following conditions: (a) The transformation from X’s to Y’s is 1-1 correspondence. (b) The function g ’s have continuous partial derivatives at all points ( x 1 ,x 2 ,...,x n ) and the n × n Jacobian determinant is non-zero, i.e. J ( x 1 ,x 2 ,...,x n ) = ± ± ± ± ± ± ± ± ± ± ± ± ± ± ∂g 1 ∂x 1 ∂g 1 ∂x 2 ··· ∂g 1 ∂x n ∂g 2 ∂x 1 ∂g 2 ∂x 2 ··· ∂g 2 ∂x n . . . . . . . . . . . . ∂g n ∂x 1 ∂g n ∂x 2 ··· ∂g n ∂x n ± ± ± ± ± ± ± ± ± ± ± ± ± ± 6 = 0 at all points ( x 1 ,x 2 ,...,x n ). Then the joint pdf of Y 1 ,Y 2 ,...,Y n is given by the following formula: f Y ( y 1 ,...,y n ) = f X ( x
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This note was uploaded on 05/04/2011 for the course STAT 1301 taught by Professor Smslee during the Spring '08 term at HKU.

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11 Example Class 9 - THE UNIVERSITY OF HONG KONG DEPARTMENT...

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