jacobian - Transformation of variables using the Jacobian...

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Unformatted text preview: Transformation of variables using the Jacobian This is an explanation of the Jacobian as W Valdar understands it. It may not be completely accurate. The general case Let X  ŸX 1 , T, X k be a k-dimensional r.v. with pdf f X Ÿx fX : Rk v R Define some 1:1 differentiable transformation of X into Y using g : R k v R k , g 1 Ÿx gŸ x  B g k Ÿx with inverse h 1 Ÿy hŸy  B h k Ÿy The pdf of Y, the transformed r.v., is f Y Ÿy  f X ŸhŸy |JŸx, y | where h 1 x 1 h 2 y 1 B h k y 1 h 1 x 2 h 2 y 2 B h k y 2 C C E C h 1 y k h 2 y k B h k y k  x1 B xk x  y1 B yk y JŸx, y  det dh dy  Ÿh 1 , T, h k Ÿy 1 , T, y k  which is often easier to calculate as J Ÿx , y  1  JŸy, x Ÿg 1 , T, g k Ÿx 1 , T, x k "1 Note: the Jacobian, J, may refer to either the determinant of the matrix (as it does here) or to the matrix of partial differentials itself. And so trivially... Let be X an r.v. with pdf f X Ÿx . Define a 1:1 strictly monotonic transformation gŸx  y, such that g : R v R, with inverse g "1 Ÿy  hŸy . Then the pdf of Y is given by f Y Ÿy  f X ŸhŸy dh dy  f X ŸhŸy dg dx "1 ...
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