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The exercise relates to joint density $$f(x,y)= Cx^{\alpha-1}y^{\beta-1}(1-x-y)^{\gamma -1}$$ for $x > 0, y > 0, x + y \leq 1$ and $$C = \frac{\Gamma(\alpha+\beta+\gamma)}{\Gamma(\alpha)\Gamma(\beta)\Gamma(\gamma)}. $$ I have already shown the marginal densities. Now I'm being asked to find the joint density of W and Z where $$W = \frac{X}{1-X},\qquad Z = \frac{Y}{1-Y}. $$ My thought is to solve $$F(w,z) = \int^\frac{x}{1-x}_{-\infty}\int^\frac{y}{1-y}_{-\infty}f(x,y) dx dy. $$ Am I on the right path? Thank you.

Did
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user42864
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  • Why not use the most powerful and simple method? – Did Nov 10 '14 at 22:24
  • Although I am sure that your method works, this complex function is far too complicated for me to perform with the technique. It has been more than 15 years since I have had to perform double integration. My calculus is extremely rusty. If you don't mind, how would I start the problem using your technique? – user42864 Nov 11 '14 at 16:51
  • Stupid me thought this was explained in details in the post I linked to. What is a specific step where you have a problem to follow the lights? – Did Nov 11 '14 at 16:58
  • Perhaps I'm making this more difficult than it actually is, but do I write both W and Z in terms of X and Y respectively and then sub for X and Y in the function? This substitution then raises the question on the limits of the integrals. – user42864 Nov 11 '14 at 17:03
  • ?? You already have "both W and Z in terms of X and Y". Sorry but unless you are much more specific about what is stopping you, I cannot help. – Did Nov 11 '14 at 17:05
  • Let me try to be clearer. Sub X with $\frac{W}{1+W}$ and sub Y with $\frac{z}{1+z}$? Then integrate $dz$ from 0 to 1 and then $dw$ from 0 to 1? – user42864 Nov 11 '14 at 17:12
  • Sorry, you seem to be making a mishmash of the very precise method I presented in the other post. Let me suggest you explain step by step how you are applying it to the present case. – Did Nov 11 '14 at 17:17
  • $E(g(v))=C\int \int g(v)[0\leq x\leq 1]\cdot \frac{x}{1-x}\cdot [0\leq y\leq 1]\frac{y}{1-y}dxdy$. Is this the correct way to set it up using your method? – user42864 Nov 11 '14 at 17:44
  • Absolutely not, you are confusing random variables and their PDFs. – Did Nov 11 '14 at 17:46
  • So X and Y both have marginal densities of a Beta function.Should I be looking at that function instead? – user42864 Nov 11 '14 at 18:32
  • I have been trying for many hours to get this problem worked out. Everything that I try with Maple comes out incorrect. So I am obviously not setting up the problem incorrectly. Could you help me out with that? I am sure that I can perform the calculations once I see what I need to do. Thank you. – user42864 Nov 16 '14 at 15:25
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    $$E(g(W,Z))=\iint g(x/(1-x),y/(1-y))f_{X,Y}(x,y)dxdy=\cdots=\iint g(w,z)f_{W,Z}(w,z)dwdz$$ – Did Nov 16 '14 at 15:28

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