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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 1 , . . . , x n ) × | J ( x 1 , . . . , x n ) | - 1 where x i = h i ( y 1 , . . . , y n ) , i = 1 , 2 , . . . , n .

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

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