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What is X1 and X2 both in terms of Y1 and Y2 if that's what I am supposed to find? I can't figure it out. How do I use the jacobian method to find the joint probability density?

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You can refer here for the methodology, and for an explicit formula for 2D case, as in yours. You have $H_1(x_1,x_2)=x_1+x_2$ and $H_2(x_1,x_2)=x_1/x_2$. You'll invert these two functions, i.e. find $x_1=H_1^{-1}(y_1,y_2),x_2=H_2^{-1}(y_1,x_2)$ and apply the formula given in the wikipedia page.

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