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Let $U$ be a uniform r.v. on $(0,1)$, define $V=1-U$. Then jointly $U$ and $V$ have flat Dirichlet distribution on a $1$-simplex. It isn't too difficult to see that the pdf is $$f(u,v)=\frac{1}{\sqrt2},\qquad(u,v)\in\mathcal{S},$$ where $\mathcal{S}=\{(u,v)\mid u,v\in[0,1],u+v=1\}$ is a simplex.

Consider now $X=a\ln U$ and $Y=b\ln V$ for some $a,b>0$, I am interested in finding the joint pdf $f(x,y)$. However, the transformation trick doesn't seem to be working due to the number of free variables. Since $U=e^{X/a}$ and $V=e^{Y/b}$, looks like the Jacobian is $$|J|=\frac{1}{ab}e^{x/a+y/b},$$ so $$f(x,y)=\frac{1}{ab\sqrt2}\exp\left(\frac{x}{a}+\frac{y}{b}\right),\qquad(x,y)\in\mathcal{T},$$ where $\mathcal{T}=\{(x,y)\mid x,y\in(-\infty,0],e^{x/a}+e^{y/b}=1\}$. Nonetheless, $f(x,y)$ does not integrate to $1$ (tested with parametrization $x=2a\ln\sin t$ and $y=2b\ln\cos t$). Where did things go wrong?

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The problem is that $U$ and $V$ do not have a joint pdf, because the random vector $(U,V)$ is concentrated on a measure zero subset of $\mathbb{R}^2$.

In particular, if we consider your candidate for the pdf, we see that $$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(u,v)\;dvdu=\frac{1}{\sqrt{2}}\int_0^1\int_{1-u}^{1-u}\;dvdu=0 $$

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