Can I identify the distribution of a random variable given a related distribution function? Let $X_1$, $X_2$ be i.i.d random variables. 
Suppose we know the distribution function of $X:=|X_1-X_2| =\max\{X_1,X_2\} - \min\{X_1,X_2\}$.
Can we find the distribution of $X_1$? 
I realize that to have any hope of identification we must fix the support of these random variables since $|X_1-X_2| = |(X_1+C)-(X_2+C)|$ for any constant $C$.
Considering first absolutely continuous random variables, with supports of the type $[a,b]$, or open ended intervals such as $[0,infinity)$, are there conditions under which the distribution of $X_1$ can be determined uniquely?
Thanks.
 A: 
Can we find the distribution of $X_1$? 

Not in general, even neglecting the translations you mention. 
To see why, assume that the distribution of $X$ is known, then $X_1-X_2$ is distributed like $UX$ where $U=\pm1$ is a symmetric Bernoulli random variable independent of $X$, hence the distribution of $X_1-X_2$ is known, that is, $|\varphi|^2$ is known, where $\varphi:t\mapsto E(\mathrm e^{\mathrm itX_1})$ is the common characteristic function of $X_1$ and $X_2$. 
Thus, to know the distribution of $X$ is equivalent to knowing $|\varphi|$. But, as is well known (!), $|\varphi|$ does not determine $\varphi$, hence  the distribution of $X$ does not determine the common distribution of $X_1$ and $X_2$.
For a counterexample, assume that $\varphi$ has period $4$ and $\varphi(t)=1-|t|$ for every $|t|\leqslant2$ (equivalently, $\varphi$ is the characteristic function of the distribution $\frac4{\pi^2}\sum\limits_n\frac1{n^2}\delta_{n\pi/2}$ where the sum is over every odd integer, but the exact form of the distribution will not be needed). Now, $\psi:t\mapsto\frac12+\frac12\varphi(2t)$ is also a characteristic function and it happens that $|\varphi|=\psi$ since $\psi$ has period $2$ and $\psi(t)=1-|t|=\varphi(t)$ for every $|t|\leqslant1$. Thus, $|\varphi|^2=|\psi|^2$ although the corresponding distributions are not translations of each other.
