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I've been given the following problem: $f(x,y) = e^{-(x+y)}$ on intervals $x \ge 0$ and $y \ge 0$, and $f(x,y) = 0$ otherwise. I'm also given that $Φ_1(x,y) = \frac{x}{y} = U$ and $Φ_2(x,y) = x + y = V$. I've proven that $f(x,y)$ is a pdf as asked by the problem but then the problem asks me to find the inverse functions of $Ψ_1(U,V) = x$ and $Ψ_2(U,V) = y$. I don't have a clue how to go about this: I certainly know how to find the inverse of a function but this looks like nothing I've seen in my textbook thus far.

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If x/y = u and x+y = v then x = uy hence uy+y = v, thus y = v/(u+1) and x = uv/(u+1). That is, Ψ1(u,v) = uv/(u+1) and Ψ2(u,v) = v/(u+1).

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  • $\begingroup$ This looks promising. Give me a second to study it. $\endgroup$ – All4KLA Oct 1 '14 at 18:12
  • $\begingroup$ Oh, I see. I was hesitant to mix the xs and ys with the us and vs hence why I was just stumped. Thank you so much! $\endgroup$ – All4KLA Oct 1 '14 at 18:14
  • $\begingroup$ Nice. I was going to ask why you had a problem with this, but you explained the reason in your comment. $\endgroup$ – Did Oct 1 '14 at 18:16
  • $\begingroup$ Yeah, it didn't look right to go about it that way but now I know. Thanks again, you're a lifesaver! $\endgroup$ – All4KLA Oct 1 '14 at 18:17

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