07jacobian - EECS 501 2-D EXAMPLE OF JACOBIAN METHOD Fall...

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EECS 501 2-D EXAMPLE OF JACOBIAN METHOD Fall 2001 Given: f x,y ( X, Y ) = 6 e - (3 X +2 Y ) for X, Y 0; 0 otherwise (2-D exponential). Goal: Compute f z,w ( Z, W ) for transformation z = x + y and w = x/ ( x + y ). 1. Compute the inverse transformation of the given one: z = z ( x, y ) = x + y w = w ( x, y ) = x/ ( x + y ) Inverse x = x ( z, w ) = zw y = y ( z, w ) = z (1 - w ) since w = x/ ( x + y ) = x/z x = zw and y = z - x = z - zw = z (1 - w ). 2. Compute the Jacobian =determinant of the Jacobian matrix J : | J | = | det ∂z ∂x ∂z ∂y ∂w ∂x ∂w ∂y | = | det 1 1 y ( x + y ) 2 - x ( x + y ) 2 | = | - 1 x + y | = 1 x + y . since x, y 0 for this particular f x,y ( X, Y ). 3. Substitute inverse transformation into f x,y ( X, Y ) / | J ( X, Y ) | : f z,w ( Z, W ) = f x,y ( X,Y ) | J ( X,Y ) | | X = x ( Z,W ) ,Y = y ( Z,W ) as defined above = f x,y ( ZW,Z (1 - W )) 1 / ( ZW + Z (1 - W )) = 6 Ze - (3 ZW +2 Z (1 - W )) = 6 Ze - Z ( W +2) for Z 0 and 0 W 1 since x, y 0 0 w 1. 4. If desired, compute marginal pdfs for z and/or w : f z ( Z ) = 6 Ze - 2 Z R 1 0 e
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  • Spring '10
  • Lehman
  • Probability theory, inverse transformation, compute marginal pdfs, Compute fz,w, Substitute inverse transformation

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