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I have a 4-dimensional joint-PDF between variables $X_1,X_2,X_3,X_4$ which are all Gaussian. I want to transform this into a 2-dimensional joint-PDF between new variables $Y_1=Y_1(X_1,X_2,X_3,X_4)$ and $Y_2=Y_2(X_1,X_2,X_3,X_4)$. I know that for a 1:1 function this is easy:

$g(\vec{Y})=f(\vec{X})\left|\frac{d\vec{X}}{d\vec{Y}}\right|$

But I simply don't see any way to do this because first of all, I have no way of finding $X_i(Y_i)$ exactly (also note that the functions $Y_i(\vec{X})$ are very long, complicated and ugly..), and even if I could, my Jacobian wouldn't be square so I could never take the determinant. I've done some looking around, and it seems as though most resources only mention how to do this for square matrices. Does anyone have any insight on how this could be done?

Thanks in advance!

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    $\begingroup$ To make it easier, what would you do if you had only two X and one Y ? Even simpler, assuming $Y = X_1 + X_2$ . In order to compute de PDF of Y, you would need to perform some integration, right ? $\endgroup$ – Florian Apr 6 at 19:29
  • $\begingroup$ Hmm. Perhaps something like this: $f_Y(Y)=\int f_{X_1X_2}(X_1,X_2) \delta(Y-(X_1+X_2))dX_1dX_2$? $\endgroup$ – zack Apr 6 at 19:36
  • $\begingroup$ Yes, I believe this is the right direction. $\endgroup$ – Florian Apr 6 at 19:41
  • $\begingroup$ Hm, so maybe for the more complex case, we'd have something like $f_{Y_1Y_2}=\int f_{X_1X_2X_3X_4}(X_1,X_2,X_3,X_4)\delta(Y_1-G_1(X_1,X_2,X_3,X_4))\delta(Y_2 - G_2(X_1,X_2,X_3,X_4)) dX_1dX_2dX_3dX_4$? $\endgroup$ – zack Apr 6 at 20:14
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    $\begingroup$ @whuber Thank you very much for your extremely detailed post at the link provided. This was extremely helpful! $\endgroup$ – zack Apr 7 at 14:42

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