Here is the problem that bugs me:
Suppose two i.i.d. random variables $X$ and $Y$, and random variable $Z$ satisfy:
$$Z = X + aY$$
where $a$ is an unknown constant.
If we happen to know the pdf of $Z$ and $X$ (and $Y$, since they are identically distributed), can we obtain the unknown constant from the given information?
Or any additional assumptions need imposing, so that $a$ can be recovered?