11.1 Example_Class_9_solution

11.1 Example_Class_9_solution - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: 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 1 ,...,x n ) × | J ( x 1 ,...,x n ) |- 1 where x i = h i ( y 1 ,...,y n ) ,i = 1 , 2 ,...,n . In fact, for easier calculation,the Jacobian determinant J ( x 1 ,x 2 ,...,x n ) can be determined: | J ( x 1 ,x 2 ,...,x n ) |- 1 = ∂h 1 ∂y 1 ∂h 1 ∂y 2 ··· ∂h 1 ∂y n ∂h 2 ∂y 1 ∂h 2 ∂y 2 ··· ∂h 2 ∂y n . . . . . . . . . . . . ∂h n ∂y 1 ∂h n ∂y 2 ··· ∂h n ∂y n 1 where h ’s are the inverse transformations x i = h i ( y 1 ,...,y n ) ,i = 1 , 2 ,...,n ....
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11.1 Example_Class_9_solution - THE UNIVERSITY OF HONG KONG...

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