Suppose I know the distribution of $Z$ and $Y$: $Z\sim F_Z$ with density $f_Z$, $Y\sim F_Y$ with density $f_Y$. Suppose I also know that $Z=X+Y$, where $X$ and $Y$ are independent and the distribution of $X$ is unknown. How to recover the distribution of $X$ and express it using $F_Z$ and $F_Y$ (or $f_Z$ and $f_Y$)?
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This is essentially deconvolution.
One approach would be to compute $\phi_X(t)$, the characteristic function of $X$ as $\phi_Z(t)/\phi_Y(t)$ (since the c.f. of $Z$ is the product of the c.f.s of $X$ and $Y$) and then from there compute the density of $X$.
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$\begingroup$ @Ben Thanks a lot, Ben! This is very helpful! One more question, after I got the characteristic function for $X$: $\phi_X(t)=\phi_Z(t)/\phi_Y(t)$, how to compute the density of $X$ using $\phi_X(t)$? $\endgroup$ Commented Jul 16, 2023 at 8:52
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1$\begingroup$ ... :^) ... In response to your question, note that the characteristic function is in effect a Fourier transform (up to a sign-flip in the exponent); it can be inverted in analogous fashion. Or you can work directly with Fourier transforms and their inverses if you prefer those, as described in the Wikipedia article on deconvolution. $\endgroup$– Glen_bCommented Jul 16, 2023 at 11:01
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1$\begingroup$ @ExcitedSnail: This is Glen's answer --- I just made a tiny correction. $\endgroup$– BenCommented Jul 16, 2023 at 11:13
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$\begingroup$ @Glen_b haha, sorry, Glen_b. I didn't read the editing history of your answer carefully. Your answer is very helpful. Will check out wikipedia deconvolution! $\endgroup$ Commented Jul 16, 2023 at 12:51