Product probability I have a random variable $Z = XY$, where $Y$ is a random variable of unknown distribution and $X$ is noise induced by a sampling process, which is uniformly distributed in $U(0,1)$. $X$ and $Y$ are independent. $Y$ is from a parametric family and non-negative.
The p.d.f. of $Z$ is given here:
$$
f_Z(z) = \int_{-\infty}^\infty f_X(x)  f_Y \left (\frac{z}{x} \right )  \frac{1}{|x|} dx,
$$
which simplifies to 
$$
f_Z(z) = \int_{0}^1 f_Y \left (\frac{z}{x} \right )  \frac{1}{|x|} dx,
$$
given $X$ is $U(0, 1)$.
First question: is the above correct and useful?
Second question: assuming I have observations for $Z$, how do I apply the formulae above to get the distribution and parameters of $Y$? In the end, I want to be able to sample from $Y$.
 A: If you observe $Z$ and assume $X$ and $Y$ are independent, then, if all you want to do is sample from $Y$, you do not even need to know the pdf of $Z$ or $Y$.
Say you observe $Z_1,...,Z_n$. Then just draw $X_1,...,X_n \sim U(0,1)$ and set $Y_i = Z_i/X_i$ for $i=1,...,n$.  
For an arbitrarily sized sample of $Y$ (one greater than $n$) you could randomly sample from $Z_1,...,Z_n$ and repeat the above.
Thanks to @whuber for pointing out that the above is incorrect.  Instead of doing that, I will propose a Method of Moments (MM) technique for the special case that $Y$ is distributed chi-squared, as suggested in the comments of the original post.
Let $Y\sim \chi (k)$ and $X \sim U(0,1)$.  Note that $E[Z]=E[XY]$.  If $cov[X,Y]=0$, which would be true under independence, then $E[XY]=E[X]E[Y]=.5 \times k$. So $2 \times E[Z]=k$ and finally;
$$
\frac{2}{n}\sum_{i=1}^n Z_i \approx k
$$
you could round the above to estimate the chi-squared parameter.  This estimate is consistent, meaning that as $n \rightarrow \infty$ it will converge on $k$.  However, it is not necessarily the most statistically efficient estimator, meaning that estimators with less variance could be proposed.  
