jacobian

# jacobian - Transformation of variables using the Jacobian...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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  X 1 , T, X k be a k-dimensional r.v. with pdf f X x fX : Rk v R Define some 1:1 differentiable transformation of X into Y using g : R k v R k , g 1 x g x  B g k x with inverse h 1 y hy  B h k y The pdf of Y, the transformed r.v., is f Y y  f X hy |Jx, 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 Jx, y  det dh dy  h 1 , T, h k y 1 , T, y k  which is often easier to calculate as J x , y  1  Jy, x g 1 , T, g k x 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 x . Define a 1:1 strictly monotonic transformation gx  y, such that g : R v R, with inverse g "1 y  hy . Then the pdf of Y is given by f Y y  f X hy dh dy  f X hy dg dx "1 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online