# Distribution of function of random variables [closed]

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 $$Y_2$$?

• What have you done so far? – gunes Mar 18 at 16:15
• 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. – Monjed Samuh Mar 18 at 16:24
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• Note that $Y_1 + 2Y_2 = (X_1 + X_2)^2$, perhaps that will be a start. – jbowman Mar 18 at 16:36
• 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. – whuber Mar 18 at 17:48