Recently I was thinking if there is a way to do the following: assuming I have some sampled points of distribution $\mathcal{X}$ and distribution $\mathcal Z$ (whose MGF I do not have in closed form) and I know that $\text{law}(\mathcal X)\cdot\text{law}(\mathcal Y) = \text{law}(\mathcal Z)$ can I obtain $\mathcal{Y}$? Are there any results/papers on this?
We can also that the support of all $3$ RVs is in $\mathbb{R\_{+}}$



Approximation to ratio distribution
$\newcommand\R{\mathbb R}\newcommand\Z{\mathbb Z}$After [James Martin's clarifying comment](https://mathoverflow.net/questions/446738/approximation-to-ratio-distribution?noredirect=1#comment1154351_446738), the question becomes as follows:



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> Suppose that $Z=XY$, where $X$ and $Y$ are independent positive random variables (r.v.'s). The distributions $P\_X$ and $P\_Z$ of $X$ and $Z$ are known. Does this determine the distribution $P\_Y$ of $Y$?
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The answer to this question is no. Indeed, write $X=e^U$, $Y=e^V$, and $Z=e^W$, where $U,V,W$ are real-valued r.v.'s such that $W=U+V$; $U$ and $V$ are independent; and the distributions $P\_U$ and $P\_W$ are known. The question can now be restated in terms of the characteristic functions (c.f.'s) $f\_U,f\_V,f\_W$ of $U,V,W$ as follows:



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> Suppose that $f\_U\,f\_V=f\_W$ and suppose that $f\_U$ and $f\_W$ are known. Does this determine $f\_V$?
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The answer to this equivalent question is of course still no. Indeed, note that (i) the function $f$ given by the formula $f(t)=\max(0,1-|t|/\pi)$ for real $t$ is a c.f. (of the absolutely continuous distribution with density $\R\ni x\mapsto\dfrac{1-\cos\pi x}{\pi^2 x^2}\,1(x\ne0)$ and (ii) the periodic function $g$ with period $2\pi$ such that $g=f$ on $[-\pi,\pi]$ is a c.f. (of the discrete distribution on $\Z$ with probability mass function
$\Z\ni x\mapsto\dfrac12\,1(x=0)+\dfrac{1-\cos\pi x}{\pi^2 x^2}\,1(x\ne0)$. Note that $fg=f^2$. So, if $f\_U=f$ and $f\_W=f^2$, then $f\_V$ can be either one of the two distinct c.f.'s: $f$ or $g$. So, $f\_U$ and $f\_W$ do not determine $f\_V$. $\quad\Box$.


This example is well known; see e.g. [this book](https://rads.stackoverflow.com/amzn/click/com/0852641702).




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Such examples are of course an exception. Indeed, if the set $N\_U:=\{t\in\R\colon f\_U(t)=0\}$ is nowhere dense (as will usually be the case, apparently -- including the case when $f\_U$ is analytic in a neighborhood of $\R$), then we know the values of $f\_V(t)=f\_W(t)/f\_U(t)$ for all $t$ in the everywhere dense set $\R\setminus N\_U$. So, by the continuity of any c.f., we know $f\_V$ completely -- so that we know the distribution of $V$ and hence the distribution of $Y$.




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Regarding concerns about statistical aspects of the problem, raised by [James Martin](https://mathoverflow.net/questions/446738/approximation-to-ratio-distribution/446854#comment1154360_446854), one can say the following.


(i) If we have an i.i.d. sample $(X\_1,Z\_1),\dots,(X\_n,Z\_n)$ from the **joint** distribution of the pair $(X,Y)$ for a large enough sample size $n$, we get the i.i.d. $Y$-sample $Y\_1:=Z\_1/X\_1,\dots,Y\_n:=Z\_n/X\_n$. So, we can use, say, the empirical c.d.f. based on $Y\_1,\dots,Y\_n$ to approximate in the standard manner the true c.d.f. of $Y$. The question as to whether the distribution of $Y$ is determined by the **individual** distributions of $X$ and and of $Z$ is completely irrelevant here, once the **joint** distribution of the pair $(X,Y)$ is known or available for sampling as described above.


(ii) If we **only** have an i.i.d. sample $X\_1,\dots,X\_n$ from the **individual distribution** of $X$ and an i.i.d. sample $Z\_1,\dots,Z\_n$ from the **individual** distribution of $Z$, then we only have partial knowledge of the **individual** distributions of $X$ and and of $Z$. But, as shown above, even complete knowledge of the **individual** distributions of $X$ and of $Z$ will in general **not** determine the distribution of $Y$. So, clearly, the partial knowledge of the **individual** distributions of $X$ and and of $Z$ cannot provide substantial partial knowledge of the distribution of $Y$ -- in particular, then we would even be unable to guess whether $Y$ is integer-valued or absolutely continuous.


Summarizing the statistical sampling aspects of the problem: depending on the kind of of sampling available -- (i) from the joint distribution of $(X,Z)$ or (ii) from the individual distributions of $X$ and $Z$, the sampling is either (i) irrelevant to the problem stated in the OP or (ii) not at all helpful to the problem.


