Why is it that if you have:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] U=g_1 (x, y), \quad V = g_2 (x,y)[/tex]

[tex] X = h_1 (u,v), \quad Y = h_2 (u,v)[/tex]

Then:

[tex]f_{U,V} (u,v) du dv = f_{X,Y} (h_1(u,v), h_2 (u,v)) \left|J(h_1(u,v),h_2(u,v))\right|^{-1} dxdy[/tex]

While when doing variable transformations in calculus, you have:

[tex]du dv = \left|J(h_1(u,v),h_2(u,v))\right| dx dy[/tex]

without the reciprocal. Why is it that with the probability densities, you take the reciprocal, rather than how it's typically done without the reciprocal?

Thanks!

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# Jacobian Transformations

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