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If two random variables $X$ and $Y$ are functions of the same one random variable $Z$, how do we find the joint PDF of $X$ and $Y$? I have seen examples where two RVs are functions of the other two RVs, but what if they are function of the same RV? Do we just "make up" a second RV, say, $V$ such that it is independent from $Z$ and has some simple distribution (it shouldn't matter I guess which one) and then proceed as in the "normal" case? But what would the inverse functions for $Z$ and $V$ be in this case? We need them for the Jacobian, right?

Also, what if there is no explicit inverse for functions $f$? For example, say $X = Z + \exp(Z)$, how do we find a PDF of $X$?

Confounded
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    In most cases there will be no PDF for $(f(Z),g(Z))$. This because mostly the image (so support of the induced distribution) will be a subset of $\mathbb R^2$ that has Lebesguemeasure $0$. If e.g. $Z$ has a distribution with support $\mathbb R$ then - if e.g. $f$ and $g$ are both the identity function t^2hen the support of $(f(Z),g(Z))=(Z,Z)$ is the set ${(z,z)\mid z\in\mathbb R}\subseteq\mathbb R$ having zero Lebesgue measure. – Vera Jul 06 '18 at 10:39
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    The last $\mathbb R$ in my former comment must be $\mathbb R^2$, sorry. – Vera Jul 06 '18 at 10:45
  • Agree with your comment, but have a further question. It will have zero measure on R^2, but if we restrict our universe to the slice where it does exist, it has non-zero measure, and is sometimes (I hesitate to say always) enumerable in terms of a modulated delta function. So why is it typically said that it doesn't exist? – John Polcari Jul 06 '18 at 12:45
  • A density wrt Lebesguemeasure on $\mathbb R^2$ does not exist and PDF's are by definition that sort of densities. A density wrt to some measure defined on subsets of ${(f(z),g(z))\mid z\in\mathbb R}\subseteq\mathbb R^2$ can exist but does not deserve the name PDF. – Vera Jul 06 '18 at 12:57
  • Perhaps, but it operates like a PDF, and deserves some name, because claiming it doesn't exist suppresses smart people from actually investigating the situation, which is how math grows. But enough philosophy - Thanks,,, – John Polcari Jul 06 '18 at 13:01
  • @Vera If PDF doesn't exist, what about a characteristic function? – Confounded Jul 09 '18 at 21:48
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    @Confounded Characteristic function always exist. The (non)existence of a PDF is not relevant. – Vera Jul 13 '18 at 18:46

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