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Let $X_1$ and $X_2$ be an iid $N(0,1)$. Suppose that $Y_1=X_1^2+X_2^2$ and $Y_2=X_1X_2$. How to find the joint pdf of $Y_1$ and $Y_2$?

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  • $\begingroup$ What have you done so far? $\endgroup$ – gunes Mar 18 at 16:15
  • $\begingroup$ I tried with transformation technique, but seems not possible because I cannot find a one-to-one transformation for X1 and X2 in terms of Y1 and Y2. $\endgroup$ – Monjed Samuh Mar 18 at 16:24
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    $\begingroup$ Welcome to CV. Since you’re new here, you may want to take our tour, which has information for new users. Since this looks like homework (apologies if it's not), please add the [self-study] tag and read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. If this is self-study rather than homework, let us know, and... it's still a good idea to show us what you've tried. $\endgroup$ – jbowman Mar 18 at 16:35
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    $\begingroup$ Note that $Y_1 + 2Y_2 = (X_1 + X_2)^2$, perhaps that will be a start. $\endgroup$ – jbowman Mar 18 at 16:36
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    $\begingroup$ Even better: $Y_1 = Z_1^2 + Z_2^2$ and $2Y_2=Z_1^2-Z_2^2$ where $Z_i=(X_1\pm X_2)/\sqrt{2}.$ Notice the $Z_i^2$ are independently distributed with $\chi^2(1)$ distributions. $\endgroup$ – whuber Mar 18 at 17:48